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 + 0.562·3-s + 4-s − 3.89·5-s − 0.562·6-s − 0.867·7-s − 8-s − 2.68·9-s + 3.89·10-s + 0.507·11-s + 0.562·12-s − 2.74·13-s + 0.867·14-s − 2.18·15-s + 16-s + 3.74·17-s + 2.68·18-s + 4.55·19-s − 3.89·20-s − 0.488·21-s − 0.507·22-s − 23-s − 0.562·24-s + 10.1·25-s + 2.74·26-s − 3.19·27-s − 0.867·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.325·3-s + 0.5·4-s − 1.73·5-s − 0.229·6-s − 0.327·7-s − 0.353·8-s − 0.894·9-s + 1.23·10-s + 0.152·11-s + 0.162·12-s − 0.760·13-s + 0.231·14-s − 0.565·15-s + 0.250·16-s + 0.908·17-s + 0.632·18-s + 1.04·19-s − 0.869·20-s − 0.106·21-s − 0.108·22-s − 0.208·23-s − 0.114·24-s + 2.02·25-s + 0.537·26-s − 0.615·27-s − 0.163·28-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 \( 1 - 0.562T + 3T^{2} \)
5 \( 1 + 3.89T + 5T^{2} \)
7 \( 1 + 0.867T + 7T^{2} \)
11 \( 1 - 0.507T + 11T^{2} \)
13 \( 1 + 2.74T + 13T^{2} \)
17 \( 1 - 3.74T + 17T^{2} \)
19 \( 1 - 4.55T + 19T^{2} \)
29 \( 1 - 5.93T + 29T^{2} \)
31 \( 1 + 1.85T + 31T^{2} \)
37 \( 1 + 0.608T + 37T^{2} \)
41 \( 1 + 9.86T + 41T^{2} \)
43 \( 1 + 1.26T + 43T^{2} \)
47 \( 1 - 8.01T + 47T^{2} \)
53 \( 1 - 4.96T + 53T^{2} \)
59 \( 1 - 5.28T + 59T^{2} \)
61 \( 1 - 13.9T + 61T^{2} \)
67 \( 1 + 3.14T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 9.43T + 73T^{2} \)
79 \( 1 + 9.73T + 79T^{2} \)
83 \( 1 + 3.20T + 83T^{2} \)
89 \( 1 - 2.82T + 89T^{2} \)
97 \( 1 - 5.75T + 97T^{2} \)
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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

−7.71258493514740288922297882356, −7.34709487106035839550943089299, −6.62186284011011733429152763571, −5.56236334580349600413859451947, −4.82851815115110899773102738580, −3.69627578565013534792409936546, −3.28134668427608410800203861060, −2.46999324053389376668107268851, −0.964329122152570059481869230684, 0, 0.964329122152570059481869230684, 2.46999324053389376668107268851, 3.28134668427608410800203861060, 3.69627578565013534792409936546, 4.82851815115110899773102738580, 5.56236334580349600413859451947, 6.62186284011011733429152763571, 7.34709487106035839550943089299, 7.71258493514740288922297882356

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