Properties

Degree 20
Conductor $ 2^{20} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s − 10·7-s − 8·9-s − 10·11-s − 10·13-s + 4·15-s + 9·17-s + 3·19-s + 20·21-s − 4·23-s − 12·25-s + 18·27-s + 13·29-s − 17·31-s + 20·33-s + 20·35-s + 11·37-s + 20·39-s + 8·41-s + 21·43-s + 16·45-s + 47-s + 55·49-s − 18·51-s + 22·53-s + 20·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 3.77·7-s − 8/3·9-s − 3.01·11-s − 2.77·13-s + 1.03·15-s + 2.18·17-s + 0.688·19-s + 4.36·21-s − 0.834·23-s − 2.39·25-s + 3.46·27-s + 2.41·29-s − 3.05·31-s + 3.48·33-s + 3.38·35-s + 1.80·37-s + 3.20·39-s + 1.24·41-s + 3.20·43-s + 2.38·45-s + 0.145·47-s + 55/7·49-s − 2.52·51-s + 3.02·53-s + 2.69·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{20} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4004} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(20,\ 2^{20} \cdot 7^{10} \cdot 11^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )$
$L(1)$  $\approx$  $3.486653432$
$L(\frac12)$  $\approx$  $3.486653432$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \(F_p\) is a polynomial of degree 20. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 19.
$p$$F_p$
bad2 \( 1 \)
7 \( ( 1 + T )^{10} \)
11 \( ( 1 + T )^{10} \)
13 \( ( 1 + T )^{10} \)
good3 \( 1 + 2 T + 4 p T^{2} + 22 T^{3} + 22 p T^{4} + 95 T^{5} + 68 p T^{6} + 166 T^{7} + 104 p T^{8} - 64 T^{9} + 304 T^{10} - 64 p T^{11} + 104 p^{3} T^{12} + 166 p^{3} T^{13} + 68 p^{5} T^{14} + 95 p^{5} T^{15} + 22 p^{7} T^{16} + 22 p^{7} T^{17} + 4 p^{9} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
5 \( 1 + 2 T + 16 T^{2} + 6 T^{3} + 96 T^{4} - 81 T^{5} + 562 T^{6} - 814 T^{7} + 2628 T^{8} - 7328 T^{9} + 386 p^{2} T^{10} - 7328 p T^{11} + 2628 p^{2} T^{12} - 814 p^{3} T^{13} + 562 p^{4} T^{14} - 81 p^{5} T^{15} + 96 p^{6} T^{16} + 6 p^{7} T^{17} + 16 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \)
17 \( 1 - 9 T + 69 T^{2} - 436 T^{3} + 2695 T^{4} - 12760 T^{5} + 58397 T^{6} - 236522 T^{7} + 993569 T^{8} - 3610763 T^{9} + 14884862 T^{10} - 3610763 p T^{11} + 993569 p^{2} T^{12} - 236522 p^{3} T^{13} + 58397 p^{4} T^{14} - 12760 p^{5} T^{15} + 2695 p^{6} T^{16} - 436 p^{7} T^{17} + 69 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \)
19 \( 1 - 3 T + 33 T^{2} - 56 T^{3} + 545 T^{4} + 40 T^{5} + 13755 T^{6} - 6782 T^{7} + 226825 T^{8} + 626813 T^{9} + 2545914 T^{10} + 626813 p T^{11} + 226825 p^{2} T^{12} - 6782 p^{3} T^{13} + 13755 p^{4} T^{14} + 40 p^{5} T^{15} + 545 p^{6} T^{16} - 56 p^{7} T^{17} + 33 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
23 \( 1 + 4 T + 45 T^{2} + 388 T^{3} + 1822 T^{4} + 9827 T^{5} + 54710 T^{6} + 168189 T^{7} + 709681 T^{8} + 2498658 T^{9} + 8288154 T^{10} + 2498658 p T^{11} + 709681 p^{2} T^{12} + 168189 p^{3} T^{13} + 54710 p^{4} T^{14} + 9827 p^{5} T^{15} + 1822 p^{6} T^{16} + 388 p^{7} T^{17} + 45 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 - 13 T + 180 T^{2} - 1277 T^{3} + 9282 T^{4} - 39834 T^{5} + 239498 T^{6} - 1116042 T^{7} + 10642989 T^{8} - 64784774 T^{9} + 453617540 T^{10} - 64784774 p T^{11} + 10642989 p^{2} T^{12} - 1116042 p^{3} T^{13} + 239498 p^{4} T^{14} - 39834 p^{5} T^{15} + 9282 p^{6} T^{16} - 1277 p^{7} T^{17} + 180 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 + 17 T + 280 T^{2} + 3041 T^{3} + 30402 T^{4} + 252539 T^{5} + 1965786 T^{6} + 13700501 T^{7} + 91046269 T^{8} + 553126762 T^{9} + 3217992092 T^{10} + 553126762 p T^{11} + 91046269 p^{2} T^{12} + 13700501 p^{3} T^{13} + 1965786 p^{4} T^{14} + 252539 p^{5} T^{15} + 30402 p^{6} T^{16} + 3041 p^{7} T^{17} + 280 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \)
37 \( 1 - 11 T + 238 T^{2} - 2392 T^{3} + 28861 T^{4} - 253174 T^{5} + 2292794 T^{6} - 17316624 T^{7} + 130729698 T^{8} - 852163815 T^{9} + 5565969504 T^{10} - 852163815 p T^{11} + 130729698 p^{2} T^{12} - 17316624 p^{3} T^{13} + 2292794 p^{4} T^{14} - 253174 p^{5} T^{15} + 28861 p^{6} T^{16} - 2392 p^{7} T^{17} + 238 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \)
41 \( 1 - 8 T + 196 T^{2} - 1079 T^{3} + 19480 T^{4} - 90187 T^{5} + 1377642 T^{6} - 5219677 T^{7} + 73916439 T^{8} - 246950713 T^{9} + 3327447044 T^{10} - 246950713 p T^{11} + 73916439 p^{2} T^{12} - 5219677 p^{3} T^{13} + 1377642 p^{4} T^{14} - 90187 p^{5} T^{15} + 19480 p^{6} T^{16} - 1079 p^{7} T^{17} + 196 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 - 21 T + 360 T^{2} - 4077 T^{3} + 936 p T^{4} - 299870 T^{5} + 1910548 T^{6} - 8211053 T^{7} + 21721750 T^{8} + 88768841 T^{9} - 948396026 T^{10} + 88768841 p T^{11} + 21721750 p^{2} T^{12} - 8211053 p^{3} T^{13} + 1910548 p^{4} T^{14} - 299870 p^{5} T^{15} + 936 p^{7} T^{16} - 4077 p^{7} T^{17} + 360 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 - T + 90 T^{2} - 205 T^{3} + 4576 T^{4} - 27410 T^{5} + 313322 T^{6} - 1926546 T^{7} + 20277487 T^{8} - 66073692 T^{9} + 1051044312 T^{10} - 66073692 p T^{11} + 20277487 p^{2} T^{12} - 1926546 p^{3} T^{13} + 313322 p^{4} T^{14} - 27410 p^{5} T^{15} + 4576 p^{6} T^{16} - 205 p^{7} T^{17} + 90 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \)
53 \( 1 - 22 T + 564 T^{2} - 8556 T^{3} + 132190 T^{4} - 1557388 T^{5} + 18086130 T^{6} - 174153460 T^{7} + 1635733868 T^{8} - 13168910514 T^{9} + 103076722766 T^{10} - 13168910514 p T^{11} + 1635733868 p^{2} T^{12} - 174153460 p^{3} T^{13} + 18086130 p^{4} T^{14} - 1557388 p^{5} T^{15} + 132190 p^{6} T^{16} - 8556 p^{7} T^{17} + 564 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \)
59 \( 1 + 24 T + 640 T^{2} + 10496 T^{3} + 168853 T^{4} + 2126939 T^{5} + 25741320 T^{6} + 263425889 T^{7} + 2583730626 T^{8} + 374242062 p T^{9} + 180948061616 T^{10} + 374242062 p^{2} T^{11} + 2583730626 p^{2} T^{12} + 263425889 p^{3} T^{13} + 25741320 p^{4} T^{14} + 2126939 p^{5} T^{15} + 168853 p^{6} T^{16} + 10496 p^{7} T^{17} + 640 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 - 16 T + 379 T^{2} - 5007 T^{3} + 75323 T^{4} - 829200 T^{5} + 9762023 T^{6} - 92713521 T^{7} + 910053001 T^{8} - 7527931174 T^{9} + 63764442150 T^{10} - 7527931174 p T^{11} + 910053001 p^{2} T^{12} - 92713521 p^{3} T^{13} + 9762023 p^{4} T^{14} - 829200 p^{5} T^{15} + 75323 p^{6} T^{16} - 5007 p^{7} T^{17} + 379 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 - 19 T + 501 T^{2} - 7052 T^{3} + 116259 T^{4} - 1338197 T^{5} + 16999939 T^{6} - 166830980 T^{7} + 1763105291 T^{8} - 14994874833 T^{9} + 135971961918 T^{10} - 14994874833 p T^{11} + 1763105291 p^{2} T^{12} - 166830980 p^{3} T^{13} + 16999939 p^{4} T^{14} - 1338197 p^{5} T^{15} + 116259 p^{6} T^{16} - 7052 p^{7} T^{17} + 501 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \)
71 \( 1 + 25 T + 571 T^{2} + 130 p T^{3} + 142268 T^{4} + 1835238 T^{5} + 22435602 T^{6} + 242676200 T^{7} + 2497305027 T^{8} + 23174712103 T^{9} + 204886130086 T^{10} + 23174712103 p T^{11} + 2497305027 p^{2} T^{12} + 242676200 p^{3} T^{13} + 22435602 p^{4} T^{14} + 1835238 p^{5} T^{15} + 142268 p^{6} T^{16} + 130 p^{8} T^{17} + 571 p^{8} T^{18} + 25 p^{9} T^{19} + p^{10} T^{20} \)
73 \( 1 - 22 T + 432 T^{2} - 79 p T^{3} + 76062 T^{4} - 850931 T^{5} + 9365994 T^{6} - 91345865 T^{7} + 882474529 T^{8} - 7774430663 T^{9} + 69193245708 T^{10} - 7774430663 p T^{11} + 882474529 p^{2} T^{12} - 91345865 p^{3} T^{13} + 9365994 p^{4} T^{14} - 850931 p^{5} T^{15} + 76062 p^{6} T^{16} - 79 p^{8} T^{17} + 432 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \)
79 \( 1 - 41 T + 1172 T^{2} - 24261 T^{3} + 420666 T^{4} - 6157896 T^{5} + 80129312 T^{6} - 931290075 T^{7} + 9950641968 T^{8} - 97876201325 T^{9} + 901731524938 T^{10} - 97876201325 p T^{11} + 9950641968 p^{2} T^{12} - 931290075 p^{3} T^{13} + 80129312 p^{4} T^{14} - 6157896 p^{5} T^{15} + 420666 p^{6} T^{16} - 24261 p^{7} T^{17} + 1172 p^{8} T^{18} - 41 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 - 8 T + 403 T^{2} - 3715 T^{3} + 86075 T^{4} - 911994 T^{5} + 12759747 T^{6} - 145050787 T^{7} + 1448607213 T^{8} - 16410880286 T^{9} + 132994013602 T^{10} - 16410880286 p T^{11} + 1448607213 p^{2} T^{12} - 145050787 p^{3} T^{13} + 12759747 p^{4} T^{14} - 911994 p^{5} T^{15} + 86075 p^{6} T^{16} - 3715 p^{7} T^{17} + 403 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
89 \( 1 - 36 T + 978 T^{2} - 18914 T^{3} + 316604 T^{4} - 4465163 T^{5} + 57229676 T^{6} - 660203752 T^{7} + 7155669276 T^{8} - 72396285400 T^{9} + 701911581402 T^{10} - 72396285400 p T^{11} + 7155669276 p^{2} T^{12} - 660203752 p^{3} T^{13} + 57229676 p^{4} T^{14} - 4465163 p^{5} T^{15} + 316604 p^{6} T^{16} - 18914 p^{7} T^{17} + 978 p^{8} T^{18} - 36 p^{9} T^{19} + p^{10} T^{20} \)
97 \( 1 + 5 T + 270 T^{2} + 560 T^{3} + 42375 T^{4} - 42794 T^{5} + 4025428 T^{6} - 28038084 T^{7} + 301112696 T^{8} - 4782252055 T^{9} + 218518076 p T^{10} - 4782252055 p T^{11} + 301112696 p^{2} T^{12} - 28038084 p^{3} T^{13} + 4025428 p^{4} T^{14} - 42794 p^{5} T^{15} + 42375 p^{6} T^{16} + 560 p^{7} T^{17} + 270 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.97832321458213724215339911632, −2.92873904115897717827762287804, −2.89844054332735198535183219260, −2.71720456969166454233443933513, −2.66588958884368118945914929282, −2.27297012882301674633631324534, −2.26179818917661492741910081567, −2.19628759009659513985509412873, −2.18343151295769478353011713357, −2.17570798896445772719305871812, −2.13055483765513823235673811653, −2.03452892552480394305352755768, −1.86491000695698988415832241966, −1.85910559747611358139286243764, −1.53047492106899188794709707974, −1.15900776653224562826512213788, −1.06693675795323359981674547799, −0.798977643183734560082492824357, −0.61380071038382942349273292698, −0.60199209395898874579751363465, −0.58896781279062181126610340364, −0.50556033794814371888545962170, −0.46658448983115109548011475678, −0.44653826345841452861847187969, −0.30440238716788046757474420969, 0.30440238716788046757474420969, 0.44653826345841452861847187969, 0.46658448983115109548011475678, 0.50556033794814371888545962170, 0.58896781279062181126610340364, 0.60199209395898874579751363465, 0.61380071038382942349273292698, 0.798977643183734560082492824357, 1.06693675795323359981674547799, 1.15900776653224562826512213788, 1.53047492106899188794709707974, 1.85910559747611358139286243764, 1.86491000695698988415832241966, 2.03452892552480394305352755768, 2.13055483765513823235673811653, 2.17570798896445772719305871812, 2.18343151295769478353011713357, 2.19628759009659513985509412873, 2.26179818917661492741910081567, 2.27297012882301674633631324534, 2.66588958884368118945914929282, 2.71720456969166454233443933513, 2.89844054332735198535183219260, 2.92873904115897717827762287804, 2.97832321458213724215339911632

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

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