Properties

Degree 2
Conductor $ 2 \cdot 23 \cdot 131 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.71·3-s + 4-s − 2.14·5-s + 1.71·6-s − 1.30·7-s − 8-s − 0.0419·9-s + 2.14·10-s + 4.74·11-s − 1.71·12-s + 1.19·13-s + 1.30·14-s + 3.68·15-s + 16-s − 1.31·17-s + 0.0419·18-s + 0.113·19-s − 2.14·20-s + 2.24·21-s − 4.74·22-s − 23-s + 1.71·24-s − 0.406·25-s − 1.19·26-s + 5.23·27-s − 1.30·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.992·3-s + 0.5·4-s − 0.958·5-s + 0.702·6-s − 0.493·7-s − 0.353·8-s − 0.0139·9-s + 0.677·10-s + 1.43·11-s − 0.496·12-s + 0.331·13-s + 0.349·14-s + 0.951·15-s + 0.250·16-s − 0.317·17-s + 0.00989·18-s + 0.0259·19-s − 0.479·20-s + 0.490·21-s − 1.01·22-s − 0.208·23-s + 0.351·24-s − 0.0812·25-s − 0.234·26-s + 1.00·27-s − 0.246·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6026\)    =    \(2 \cdot 23 \cdot 131\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6026} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6026,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;23,\;131\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;23,\;131\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
23 \( 1 + T \)
131 \( 1 + T \)
good3 \( 1 + 1.71T + 3T^{2} \)
5 \( 1 + 2.14T + 5T^{2} \)
7 \( 1 + 1.30T + 7T^{2} \)
11 \( 1 - 4.74T + 11T^{2} \)
13 \( 1 - 1.19T + 13T^{2} \)
17 \( 1 + 1.31T + 17T^{2} \)
19 \( 1 - 0.113T + 19T^{2} \)
29 \( 1 + 8.62T + 29T^{2} \)
31 \( 1 + 1.19T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 - 4.88T + 41T^{2} \)
43 \( 1 - 6.12T + 43T^{2} \)
47 \( 1 - 7.88T + 47T^{2} \)
53 \( 1 - 4.06T + 53T^{2} \)
59 \( 1 + 9.74T + 59T^{2} \)
61 \( 1 - 0.336T + 61T^{2} \)
67 \( 1 - 8.95T + 67T^{2} \)
71 \( 1 + 1.24T + 71T^{2} \)
73 \( 1 + 9.56T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 - 5.53T + 89T^{2} \)
97 \( 1 - 8.36T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.54744555127983959049006569834, −7.13511411838575378079200988883, −6.20998557342261472072304033866, −5.96271258449665971689218361822, −4.86855804390999418596140386076, −3.89157047286246840495638391395, −3.44248949996561541840873634296, −2.07481779835182332470161065164, −0.914356240714065580847944794014, 0, 0.914356240714065580847944794014, 2.07481779835182332470161065164, 3.44248949996561541840873634296, 3.89157047286246840495638391395, 4.86855804390999418596140386076, 5.96271258449665971689218361822, 6.20998557342261472072304033866, 7.13511411838575378079200988883, 7.54744555127983959049006569834

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