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 + 0.180·3-s + 4-s − 3.11·5-s − 0.180·6-s + 2.83·7-s − 8-s − 2.96·9-s + 3.11·10-s + 1.84·11-s + 0.180·12-s − 3.19·13-s − 2.83·14-s − 0.561·15-s + 16-s + 5.26·17-s + 2.96·18-s − 2.30·19-s − 3.11·20-s + 0.510·21-s − 1.84·22-s + 23-s − 0.180·24-s + 4.71·25-s + 3.19·26-s − 1.07·27-s + 2.83·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.103·3-s + 0.5·4-s − 1.39·5-s − 0.0735·6-s + 1.07·7-s − 0.353·8-s − 0.989·9-s + 0.985·10-s + 0.555·11-s + 0.0519·12-s − 0.887·13-s − 0.758·14-s − 0.144·15-s + 0.250·16-s + 1.27·17-s + 0.699·18-s − 0.529·19-s − 0.696·20-s + 0.111·21-s − 0.392·22-s + 0.208·23-s − 0.0367·24-s + 0.942·25-s + 0.627·26-s − 0.206·27-s + 0.536·28-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 \( 1 - 0.180T + 3T^{2} \)
5 \( 1 + 3.11T + 5T^{2} \)
7 \( 1 - 2.83T + 7T^{2} \)
11 \( 1 - 1.84T + 11T^{2} \)
13 \( 1 + 3.19T + 13T^{2} \)
17 \( 1 - 5.26T + 17T^{2} \)
19 \( 1 + 2.30T + 19T^{2} \)
29 \( 1 + 4.19T + 29T^{2} \)
31 \( 1 - 0.161T + 31T^{2} \)
37 \( 1 - 3.37T + 37T^{2} \)
41 \( 1 - 7.27T + 41T^{2} \)
43 \( 1 + 4.10T + 43T^{2} \)
47 \( 1 - 8.67T + 47T^{2} \)
53 \( 1 + 5.41T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 4.07T + 61T^{2} \)
67 \( 1 + 8.82T + 67T^{2} \)
71 \( 1 - 3.83T + 71T^{2} \)
73 \( 1 + 3.38T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 - 4.17T + 89T^{2} \)
97 \( 1 + 0.272T + 97T^{2} \)
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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

−7.81164111151874966877908445004, −7.47270357328666368683671747028, −6.46391934268394622184156136782, −5.56992246937627775351162803536, −4.79953379892272203046334183195, −3.96969523970249728078718261392, −3.18747850475691269886225674568, −2.26456493095544430721027210430, −1.11104974692636494669543908376, 0, 1.11104974692636494669543908376, 2.26456493095544430721027210430, 3.18747850475691269886225674568, 3.96969523970249728078718261392, 4.79953379892272203046334183195, 5.56992246937627775351162803536, 6.46391934268394622184156136782, 7.47270357328666368683671747028, 7.81164111151874966877908445004

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