Properties

Label 32-42e32-1.1-c1e16-0-0
Degree $32$
Conductor $8.790\times 10^{51}$
Sign $1$
Analytic cond. $2.40110\times 10^{18}$
Root an. cond. $3.75308$
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 + 30·16-s + 48·25-s + 72·64-s + 384·100-s + 96·109-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 192·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 4·4-s + 15/2·16-s + 48/5·25-s + 9·64-s + 38.3·100-s + 9.19·109-s − 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 14.7·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(2.40110\times 10^{18}\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.04319565321\)
\(L(\frac12)\) \(\approx\) \(0.04319565321\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
11 \( ( 1 + 8 T^{2} + 6 p T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
13 \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{8} \)
17 \( ( 1 - 12 T^{2} + 506 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
19 \( ( 1 - 28 T^{2} + 870 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
23 \( ( 1 + 56 T^{2} + 1650 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
29 \( ( 1 - 88 T^{2} + 3426 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
31 \( ( 1 + 20 T^{2} + 1590 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
37 \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{8} \)
41 \( ( 1 - 60 T^{2} + 3674 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
43 \( ( 1 - 76 T^{2} + 4950 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
47 \( ( 1 + 68 T^{2} + 2502 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
53 \( ( 1 - 128 T^{2} + 7986 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
59 \( ( 1 + 20 T^{2} + 150 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
61 \( ( 1 - 72 T^{2} + 3938 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
67 \( ( 1 - 196 T^{2} + 17814 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
71 \( ( 1 + 8 T^{2} + 7026 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
73 \( ( 1 + 120 T^{2} + 7346 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
79 \( ( 1 - 100 T^{2} + 8070 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
83 \( ( 1 + 44 T^{2} + 12534 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
89 \( ( 1 - 252 T^{2} + 31130 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
97 \( ( 1 + 72 T^{2} + 15314 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
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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

−2.29998349022199382262608128622, −2.27682443552204832855126895819, −2.24958134113888676503354150447, −2.24376492406876774383590876955, −2.01767613453965291186470089472, −2.00138349975699348472149359525, −2.00095463205326088658225907734, −1.86556660497002907852933534240, −1.73707827368119040034246332322, −1.69937797511483983292399138004, −1.59528905667869107886989077524, −1.40937871500138743828337130498, −1.35631345910849485935626352593, −1.20232809650163689749947458703, −1.19239835061893746890998074534, −1.09318554584262477602384332641, −1.07481071144457209949236145061, −1.07353162861505643655201912527, −1.06389229011593480632931069136, −0.805491881905255210414245545849, −0.804631926316138298542881284945, −0.54860424030202857223233173013, −0.44588759378127148790559080637, −0.33478897416353256759110442435, −0.00276940359937714551952131384, 0.00276940359937714551952131384, 0.33478897416353256759110442435, 0.44588759378127148790559080637, 0.54860424030202857223233173013, 0.804631926316138298542881284945, 0.805491881905255210414245545849, 1.06389229011593480632931069136, 1.07353162861505643655201912527, 1.07481071144457209949236145061, 1.09318554584262477602384332641, 1.19239835061893746890998074534, 1.20232809650163689749947458703, 1.35631345910849485935626352593, 1.40937871500138743828337130498, 1.59528905667869107886989077524, 1.69937797511483983292399138004, 1.73707827368119040034246332322, 1.86556660497002907852933534240, 2.00095463205326088658225907734, 2.00138349975699348472149359525, 2.01767613453965291186470089472, 2.24376492406876774383590876955, 2.24958134113888676503354150447, 2.27682443552204832855126895819, 2.29998349022199382262608128622

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

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