Properties

Label 32-33e32-1.1-c1e16-0-1
Degree $32$
Conductor $3.912\times 10^{48}$
Sign $1$
Analytic cond. $1.06875\times 10^{15}$
Root an. cond. $2.94884$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s + 34·16-s + 24·25-s + 16·31-s + 8·37-s + 64·49-s − 132·64-s + 96·67-s − 192·100-s − 64·103-s − 128·124-s + 127-s + 131-s + 137-s + 139-s − 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 136·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 4·4-s + 17/2·16-s + 24/5·25-s + 2.87·31-s + 1.31·37-s + 64/7·49-s − 16.5·64-s + 11.7·67-s − 19.1·100-s − 6.30·103-s − 11.4·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 10.4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 11^{32}\)
Sign: $1$
Analytic conductor: \(1.06875\times 10^{15}\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1089} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 11^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12.08879977\)
\(L(\frac12)\) \(\approx\) \(12.08879977\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( ( 1 + p^{2} T^{2} + 7 T^{4} + 11 p T^{6} + 65 T^{8} + 11 p^{3} T^{10} + 7 p^{4} T^{12} + p^{8} T^{14} + p^{8} T^{16} )^{2} \)
5 \( ( 1 - 12 T^{2} + 99 T^{4} - 658 T^{6} + 3381 T^{8} - 658 p^{2} T^{10} + 99 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
7 \( ( 1 - 32 T^{2} + 10 p^{2} T^{4} - 5072 T^{6} + 40139 T^{8} - 5072 p^{2} T^{10} + 10 p^{6} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 - 68 T^{2} + 2290 T^{4} - 49568 T^{6} + 757259 T^{8} - 49568 p^{2} T^{10} + 2290 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( ( 1 + 86 T^{2} + 3807 T^{4} + 109498 T^{6} + 2200505 T^{8} + 109498 p^{2} T^{10} + 3807 p^{4} T^{12} + 86 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - 86 T^{2} + 3835 T^{4} - 114044 T^{6} + 2498549 T^{8} - 114044 p^{2} T^{10} + 3835 p^{4} T^{12} - 86 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 102 T^{2} + 5430 T^{4} - 195472 T^{6} + 5177079 T^{8} - 195472 p^{2} T^{10} + 5430 p^{4} T^{12} - 102 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 + 100 T^{2} + 4624 T^{4} + 139900 T^{6} + 3859406 T^{8} + 139900 p^{2} T^{10} + 4624 p^{4} T^{12} + 100 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 4 T + 95 T^{2} - 216 T^{3} + 3799 T^{4} - 216 p T^{5} + 95 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
37 \( ( 1 - 2 T + 72 T^{2} - 120 T^{3} + 3841 T^{4} - 120 p T^{5} + 72 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
41 \( ( 1 + 86 T^{2} + 7470 T^{4} + 424744 T^{6} + 19373639 T^{8} + 424744 p^{2} T^{10} + 7470 p^{4} T^{12} + 86 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 180 T^{2} + 12786 T^{4} - 463360 T^{6} + 14361051 T^{8} - 463360 p^{2} T^{10} + 12786 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 - 154 T^{2} + 13707 T^{4} - 891482 T^{6} + 45460025 T^{8} - 891482 p^{2} T^{10} + 13707 p^{4} T^{12} - 154 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 202 T^{2} + 23015 T^{4} - 1862022 T^{6} + 112849729 T^{8} - 1862022 p^{2} T^{10} + 23015 p^{4} T^{12} - 202 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 - 420 T^{2} + 79419 T^{4} - 150350 p T^{6} + 642105261 T^{8} - 150350 p^{3} T^{10} + 79419 p^{4} T^{12} - 420 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 302 T^{2} + 46183 T^{4} - 4646264 T^{6} + 5459105 p T^{8} - 4646264 p^{2} T^{10} + 46183 p^{4} T^{12} - 302 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 24 T + 429 T^{2} - 5008 T^{3} + 47859 T^{4} - 5008 p T^{5} + 429 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
71 \( ( 1 - 436 T^{2} + 87795 T^{4} - 10860674 T^{6} + 918360869 T^{8} - 10860674 p^{2} T^{10} + 87795 p^{4} T^{12} - 436 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 244 T^{2} + 32352 T^{4} - 3391772 T^{6} + 284923310 T^{8} - 3391772 p^{2} T^{10} + 32352 p^{4} T^{12} - 244 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 452 T^{2} + 97138 T^{4} - 13137344 T^{6} + 1231054955 T^{8} - 13137344 p^{2} T^{10} + 97138 p^{4} T^{12} - 452 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( ( 1 + 386 T^{2} + 81822 T^{4} + 11270608 T^{6} + 1105204655 T^{8} + 11270608 p^{2} T^{10} + 81822 p^{4} T^{12} + 386 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 - 582 T^{2} + 157158 T^{4} - 25689904 T^{6} + 2778331335 T^{8} - 25689904 p^{2} T^{10} + 157158 p^{4} T^{12} - 582 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 + 133 T^{2} - 330 T^{3} + 8239 T^{4} - 330 p T^{5} + 133 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.60151406778585176700772035822, −2.58868617198029418807404189560, −2.56028238562377240262453770252, −2.41893417288374219102995254190, −2.24001688825321628432367146531, −2.22910118221352193450796195025, −2.11557891515205814980289649053, −2.07359307152150981892691724188, −1.95022436284174280585114856610, −1.85122159132091798126713128341, −1.76134495542290762331392165882, −1.38757055763378197712996093896, −1.30492980399026857215308234693, −1.23587956172885078500293519547, −1.20984153297387632775226526441, −1.18647733965341562787112141491, −1.11819915132474521782161659814, −1.11118774541360845615008150573, −0.968933882180737361786210759237, −0.74272296545633417627331677603, −0.66252041845139694383057129592, −0.52651304948994397670357587418, −0.51408885130907543948359990438, −0.41142906882889670170252438146, −0.23600968729111307105706260439, 0.23600968729111307105706260439, 0.41142906882889670170252438146, 0.51408885130907543948359990438, 0.52651304948994397670357587418, 0.66252041845139694383057129592, 0.74272296545633417627331677603, 0.968933882180737361786210759237, 1.11118774541360845615008150573, 1.11819915132474521782161659814, 1.18647733965341562787112141491, 1.20984153297387632775226526441, 1.23587956172885078500293519547, 1.30492980399026857215308234693, 1.38757055763378197712996093896, 1.76134495542290762331392165882, 1.85122159132091798126713128341, 1.95022436284174280585114856610, 2.07359307152150981892691724188, 2.11557891515205814980289649053, 2.22910118221352193450796195025, 2.24001688825321628432367146531, 2.41893417288374219102995254190, 2.56028238562377240262453770252, 2.58868617198029418807404189560, 2.60151406778585176700772035822

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

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