Properties

Degree 22
Conductor $ 2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 43^{11} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 11·2-s − 3-s + 66·4-s − 11·5-s + 11·6-s + 6·7-s − 286·8-s − 10·9-s + 121·10-s − 11·11-s − 66·12-s + 4·13-s − 66·14-s + 11·15-s + 1.00e3·16-s − 10·17-s + 110·18-s + 14·19-s − 726·20-s − 6·21-s + 121·22-s − 18·23-s + 286·24-s + 66·25-s − 44·26-s + 12·27-s + 396·28-s + ⋯
L(s)  = 1  − 7.77·2-s − 0.577·3-s + 33·4-s − 4.91·5-s + 4.49·6-s + 2.26·7-s − 101.·8-s − 3.33·9-s + 38.2·10-s − 3.31·11-s − 19.0·12-s + 1.10·13-s − 17.6·14-s + 2.84·15-s + 250.·16-s − 2.42·17-s + 25.9·18-s + 3.21·19-s − 162.·20-s − 1.30·21-s + 25.7·22-s − 3.75·23-s + 58.3·24-s + 66/5·25-s − 8.62·26-s + 2.30·27-s + 74.8·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 43^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 43^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(22\)
\( N \)  =  \(2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 43^{11}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4730} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(22,\ 2^{11} \cdot 5^{11} \cdot 11^{11} \cdot 43^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )$
$L(1)$  $\approx$  $0.008286555656$
$L(\frac12)$  $\approx$  $0.008286555656$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\(F_p(T)\) is a polynomial of degree 22. If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 21.
$p$$F_p(T)$
bad2 \( ( 1 + T )^{11} \)
5 \( ( 1 + T )^{11} \)
11 \( ( 1 + T )^{11} \)
43 \( ( 1 - T )^{11} \)
good3 \( 1 + T + 11 T^{2} + p^{2} T^{3} + 22 p T^{4} + 31 T^{5} + 257 T^{6} - 35 T^{7} + 704 T^{8} - 725 T^{9} + 182 p^{2} T^{10} - 3146 T^{11} + 182 p^{3} T^{12} - 725 p^{2} T^{13} + 704 p^{3} T^{14} - 35 p^{4} T^{15} + 257 p^{5} T^{16} + 31 p^{6} T^{17} + 22 p^{8} T^{18} + p^{10} T^{19} + 11 p^{9} T^{20} + p^{10} T^{21} + p^{11} T^{22} \)
7 \( 1 - 6 T + 41 T^{2} - 195 T^{3} + 853 T^{4} - 456 p T^{5} + 11332 T^{6} - 5044 p T^{7} + 109189 T^{8} - 304757 T^{9} + 854564 T^{10} - 2261356 T^{11} + 854564 p T^{12} - 304757 p^{2} T^{13} + 109189 p^{3} T^{14} - 5044 p^{5} T^{15} + 11332 p^{5} T^{16} - 456 p^{7} T^{17} + 853 p^{7} T^{18} - 195 p^{8} T^{19} + 41 p^{9} T^{20} - 6 p^{10} T^{21} + p^{11} T^{22} \)
13 \( 1 - 4 T + 69 T^{2} - 214 T^{3} + 2317 T^{4} - 6388 T^{5} + 55721 T^{6} - 144536 T^{7} + 1064202 T^{8} - 2539080 T^{9} + 16477738 T^{10} - 35978532 T^{11} + 16477738 p T^{12} - 2539080 p^{2} T^{13} + 1064202 p^{3} T^{14} - 144536 p^{4} T^{15} + 55721 p^{5} T^{16} - 6388 p^{6} T^{17} + 2317 p^{7} T^{18} - 214 p^{8} T^{19} + 69 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \)
17 \( 1 + 10 T + 113 T^{2} + 787 T^{3} + 5635 T^{4} + 31710 T^{5} + 178382 T^{6} + 861634 T^{7} + 4228065 T^{8} + 18442633 T^{9} + 82974330 T^{10} + 335085016 T^{11} + 82974330 p T^{12} + 18442633 p^{2} T^{13} + 4228065 p^{3} T^{14} + 861634 p^{4} T^{15} + 178382 p^{5} T^{16} + 31710 p^{6} T^{17} + 5635 p^{7} T^{18} + 787 p^{8} T^{19} + 113 p^{9} T^{20} + 10 p^{10} T^{21} + p^{11} T^{22} \)
19 \( 1 - 14 T + 139 T^{2} - 1081 T^{3} + 7803 T^{4} - 51898 T^{5} + 318510 T^{6} - 1798606 T^{7} + 9535247 T^{8} - 47724665 T^{9} + 226979392 T^{10} - 1016995472 T^{11} + 226979392 p T^{12} - 47724665 p^{2} T^{13} + 9535247 p^{3} T^{14} - 1798606 p^{4} T^{15} + 318510 p^{5} T^{16} - 51898 p^{6} T^{17} + 7803 p^{7} T^{18} - 1081 p^{8} T^{19} + 139 p^{9} T^{20} - 14 p^{10} T^{21} + p^{11} T^{22} \)
23 \( 1 + 18 T + 319 T^{2} + 3714 T^{3} + 40117 T^{4} + 351722 T^{5} + 2862803 T^{6} + 878472 p T^{7} + 133116154 T^{8} + 781361156 T^{9} + 4297024446 T^{10} + 21277738828 T^{11} + 4297024446 p T^{12} + 781361156 p^{2} T^{13} + 133116154 p^{3} T^{14} + 878472 p^{5} T^{15} + 2862803 p^{5} T^{16} + 351722 p^{6} T^{17} + 40117 p^{7} T^{18} + 3714 p^{8} T^{19} + 319 p^{9} T^{20} + 18 p^{10} T^{21} + p^{11} T^{22} \)
29 \( 1 + 6 T + 185 T^{2} + 1098 T^{3} + 17276 T^{4} + 100744 T^{5} + 1083130 T^{6} + 6059792 T^{7} + 50753057 T^{8} + 264521298 T^{9} + 1857219847 T^{10} + 8754314412 T^{11} + 1857219847 p T^{12} + 264521298 p^{2} T^{13} + 50753057 p^{3} T^{14} + 6059792 p^{4} T^{15} + 1083130 p^{5} T^{16} + 100744 p^{6} T^{17} + 17276 p^{7} T^{18} + 1098 p^{8} T^{19} + 185 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \)
31 \( 1 - 13 T + 316 T^{2} - 3389 T^{3} + 45996 T^{4} - 414137 T^{5} + 4077083 T^{6} - 31182876 T^{7} + 244722226 T^{8} - 1600202938 T^{9} + 10449201896 T^{10} - 58438954190 T^{11} + 10449201896 p T^{12} - 1600202938 p^{2} T^{13} + 244722226 p^{3} T^{14} - 31182876 p^{4} T^{15} + 4077083 p^{5} T^{16} - 414137 p^{6} T^{17} + 45996 p^{7} T^{18} - 3389 p^{8} T^{19} + 316 p^{9} T^{20} - 13 p^{10} T^{21} + p^{11} T^{22} \)
37 \( 1 - T + 186 T^{2} + 69 T^{3} + 18790 T^{4} + 19123 T^{5} + 1371913 T^{6} + 1719148 T^{7} + 77809346 T^{8} + 102179878 T^{9} + 3522899448 T^{10} + 4450627678 T^{11} + 3522899448 p T^{12} + 102179878 p^{2} T^{13} + 77809346 p^{3} T^{14} + 1719148 p^{4} T^{15} + 1371913 p^{5} T^{16} + 19123 p^{6} T^{17} + 18790 p^{7} T^{18} + 69 p^{8} T^{19} + 186 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \)
41 \( 1 - 12 T + 281 T^{2} - 2910 T^{3} + 39921 T^{4} - 367844 T^{5} + 3783993 T^{6} - 31007864 T^{7} + 261909310 T^{8} - 1908124592 T^{9} + 13837625822 T^{10} - 89265217428 T^{11} + 13837625822 p T^{12} - 1908124592 p^{2} T^{13} + 261909310 p^{3} T^{14} - 31007864 p^{4} T^{15} + 3783993 p^{5} T^{16} - 367844 p^{6} T^{17} + 39921 p^{7} T^{18} - 2910 p^{8} T^{19} + 281 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \)
47 \( 1 + 5 T + 211 T^{2} + 1191 T^{3} + 26133 T^{4} + 146594 T^{5} + 2359788 T^{6} + 12684950 T^{7} + 166644597 T^{8} + 834900889 T^{9} + 9555736092 T^{10} + 43627217926 T^{11} + 9555736092 p T^{12} + 834900889 p^{2} T^{13} + 166644597 p^{3} T^{14} + 12684950 p^{4} T^{15} + 2359788 p^{5} T^{16} + 146594 p^{6} T^{17} + 26133 p^{7} T^{18} + 1191 p^{8} T^{19} + 211 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \)
53 \( 1 + 27 T + 683 T^{2} + 11403 T^{3} + 176440 T^{4} + 2209291 T^{5} + 26016523 T^{6} + 265446321 T^{7} + 2573710310 T^{8} + 22256645035 T^{9} + 183789838266 T^{10} + 1368384439916 T^{11} + 183789838266 p T^{12} + 22256645035 p^{2} T^{13} + 2573710310 p^{3} T^{14} + 265446321 p^{4} T^{15} + 26016523 p^{5} T^{16} + 2209291 p^{6} T^{17} + 176440 p^{7} T^{18} + 11403 p^{8} T^{19} + 683 p^{9} T^{20} + 27 p^{10} T^{21} + p^{11} T^{22} \)
59 \( 1 - 11 T + 467 T^{2} - 3647 T^{3} + 89683 T^{4} - 438686 T^{5} + 9084162 T^{6} - 13845946 T^{7} + 519801447 T^{8} + 1822460995 T^{9} + 19762126954 T^{10} + 203907662094 T^{11} + 19762126954 p T^{12} + 1822460995 p^{2} T^{13} + 519801447 p^{3} T^{14} - 13845946 p^{4} T^{15} + 9084162 p^{5} T^{16} - 438686 p^{6} T^{17} + 89683 p^{7} T^{18} - 3647 p^{8} T^{19} + 467 p^{9} T^{20} - 11 p^{10} T^{21} + p^{11} T^{22} \)
61 \( 1 - 36 T + 839 T^{2} - 14612 T^{3} + 218860 T^{4} - 2882022 T^{5} + 34433520 T^{6} - 373718240 T^{7} + 3744303379 T^{8} - 34749872166 T^{9} + 301335794361 T^{10} - 2434628504696 T^{11} + 301335794361 p T^{12} - 34749872166 p^{2} T^{13} + 3744303379 p^{3} T^{14} - 373718240 p^{4} T^{15} + 34433520 p^{5} T^{16} - 2882022 p^{6} T^{17} + 218860 p^{7} T^{18} - 14612 p^{8} T^{19} + 839 p^{9} T^{20} - 36 p^{10} T^{21} + p^{11} T^{22} \)
67 \( 1 - 18 T + 557 T^{2} - 7280 T^{3} + 133099 T^{4} - 1446074 T^{5} + 20250607 T^{6} - 194050496 T^{7} + 2253964178 T^{8} - 19229019076 T^{9} + 192042617706 T^{10} - 1459223924640 T^{11} + 192042617706 p T^{12} - 19229019076 p^{2} T^{13} + 2253964178 p^{3} T^{14} - 194050496 p^{4} T^{15} + 20250607 p^{5} T^{16} - 1446074 p^{6} T^{17} + 133099 p^{7} T^{18} - 7280 p^{8} T^{19} + 557 p^{9} T^{20} - 18 p^{10} T^{21} + p^{11} T^{22} \)
71 \( 1 + 14 T + 437 T^{2} + 4223 T^{3} + 71293 T^{4} + 481786 T^{5} + 6090758 T^{6} + 32381282 T^{7} + 417026685 T^{8} + 2611708381 T^{9} + 33828315852 T^{10} + 225169195076 T^{11} + 33828315852 p T^{12} + 2611708381 p^{2} T^{13} + 417026685 p^{3} T^{14} + 32381282 p^{4} T^{15} + 6090758 p^{5} T^{16} + 481786 p^{6} T^{17} + 71293 p^{7} T^{18} + 4223 p^{8} T^{19} + 437 p^{9} T^{20} + 14 p^{10} T^{21} + p^{11} T^{22} \)
73 \( 1 + 11 T + 419 T^{2} + 4678 T^{3} + 89099 T^{4} + 977295 T^{5} + 12973961 T^{6} + 132506040 T^{7} + 1444417742 T^{8} + 13285903126 T^{9} + 129124939882 T^{10} + 1067800677380 T^{11} + 129124939882 p T^{12} + 13285903126 p^{2} T^{13} + 1444417742 p^{3} T^{14} + 132506040 p^{4} T^{15} + 12973961 p^{5} T^{16} + 977295 p^{6} T^{17} + 89099 p^{7} T^{18} + 4678 p^{8} T^{19} + 419 p^{9} T^{20} + 11 p^{10} T^{21} + p^{11} T^{22} \)
79 \( 1 - 28 T + 932 T^{2} - 18511 T^{3} + 366453 T^{4} - 5675049 T^{5} + 84285456 T^{6} - 1068914179 T^{7} + 12909518055 T^{8} - 137778481749 T^{9} + 1401316366039 T^{10} - 12753157348064 T^{11} + 1401316366039 p T^{12} - 137778481749 p^{2} T^{13} + 12909518055 p^{3} T^{14} - 1068914179 p^{4} T^{15} + 84285456 p^{5} T^{16} - 5675049 p^{6} T^{17} + 366453 p^{7} T^{18} - 18511 p^{8} T^{19} + 932 p^{9} T^{20} - 28 p^{10} T^{21} + p^{11} T^{22} \)
83 \( 1 + 4 T + 376 T^{2} + 1039 T^{3} + 71899 T^{4} + 183149 T^{5} + 10363006 T^{6} + 29383751 T^{7} + 1227190655 T^{8} + 3388895417 T^{9} + 119250890355 T^{10} + 299932335704 T^{11} + 119250890355 p T^{12} + 3388895417 p^{2} T^{13} + 1227190655 p^{3} T^{14} + 29383751 p^{4} T^{15} + 10363006 p^{5} T^{16} + 183149 p^{6} T^{17} + 71899 p^{7} T^{18} + 1039 p^{8} T^{19} + 376 p^{9} T^{20} + 4 p^{10} T^{21} + p^{11} T^{22} \)
89 \( 1 + 7 T + 582 T^{2} + 4287 T^{3} + 168480 T^{4} + 1270715 T^{5} + 32203935 T^{6} + 241593516 T^{7} + 4553415280 T^{8} + 32830868478 T^{9} + 504268929186 T^{10} + 3348280634826 T^{11} + 504268929186 p T^{12} + 32830868478 p^{2} T^{13} + 4553415280 p^{3} T^{14} + 241593516 p^{4} T^{15} + 32203935 p^{5} T^{16} + 1270715 p^{6} T^{17} + 168480 p^{7} T^{18} + 4287 p^{8} T^{19} + 582 p^{9} T^{20} + 7 p^{10} T^{21} + p^{11} T^{22} \)
97 \( 1 + T + 387 T^{2} + 1342 T^{3} + 82555 T^{4} + 470349 T^{5} + 13051513 T^{6} + 93397112 T^{7} + 1738013254 T^{8} + 13081357618 T^{9} + 198916145922 T^{10} + 1415625834516 T^{11} + 198916145922 p T^{12} + 13081357618 p^{2} T^{13} + 1738013254 p^{3} T^{14} + 93397112 p^{4} T^{15} + 13051513 p^{5} T^{16} + 470349 p^{6} T^{17} + 82555 p^{7} T^{18} + 1342 p^{8} T^{19} + 387 p^{9} T^{20} + p^{10} T^{21} + p^{11} T^{22} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.73325079761686572739377972141, −2.64605146804612419610100810680, −2.36266546166539410307706445465, −2.14709898995861867837113293022, −2.01649473733527147337702941008, −1.96356750447216182688987730020, −1.96166645436441330337968215608, −1.93845727174227136478418400532, −1.92939681618810558623338213750, −1.86845399323764907699654322076, −1.72531420680418500724607370469, −1.48157224347633364163675740553, −1.42006492374519357981218225615, −1.36120752272628571638989939784, −1.07556726665076332465419857810, −0.919206797848796634323582972701, −0.863526906766281914854698705206, −0.828332573334599911834531823611, −0.60428133488803621536962010390, −0.53627798208285717222032080721, −0.52671822966907940508622333913, −0.44466346430493314433545081877, −0.38912143377736055310545186550, −0.33903209887734812572239508407, −0.07406601144925026875421563206, 0.07406601144925026875421563206, 0.33903209887734812572239508407, 0.38912143377736055310545186550, 0.44466346430493314433545081877, 0.52671822966907940508622333913, 0.53627798208285717222032080721, 0.60428133488803621536962010390, 0.828332573334599911834531823611, 0.863526906766281914854698705206, 0.919206797848796634323582972701, 1.07556726665076332465419857810, 1.36120752272628571638989939784, 1.42006492374519357981218225615, 1.48157224347633364163675740553, 1.72531420680418500724607370469, 1.86845399323764907699654322076, 1.92939681618810558623338213750, 1.93845727174227136478418400532, 1.96166645436441330337968215608, 1.96356750447216182688987730020, 2.01649473733527147337702941008, 2.14709898995861867837113293022, 2.36266546166539410307706445465, 2.64605146804612419610100810680, 2.73325079761686572739377972141

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.