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 − 2.94·3-s + 4-s + 2.28·5-s + 2.94·6-s − 4.71·7-s − 8-s + 5.65·9-s − 2.28·10-s + 1.43·11-s − 2.94·12-s + 1.73·13-s + 4.71·14-s − 6.71·15-s + 16-s + 7.47·17-s − 5.65·18-s − 7.14·19-s + 2.28·20-s + 13.8·21-s − 1.43·22-s + 23-s + 2.94·24-s + 0.212·25-s − 1.73·26-s − 7.81·27-s − 4.71·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.69·3-s + 0.5·4-s + 1.02·5-s + 1.20·6-s − 1.78·7-s − 0.353·8-s + 1.88·9-s − 0.721·10-s + 0.433·11-s − 0.849·12-s + 0.480·13-s + 1.25·14-s − 1.73·15-s + 0.250·16-s + 1.81·17-s − 1.33·18-s − 1.63·19-s + 0.510·20-s + 3.02·21-s − 0.306·22-s + 0.208·23-s + 0.600·24-s + 0.0425·25-s − 0.339·26-s − 1.50·27-s − 0.890·28-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 \( 1 + 2.94T + 3T^{2} \)
5 \( 1 - 2.28T + 5T^{2} \)
7 \( 1 + 4.71T + 7T^{2} \)
11 \( 1 - 1.43T + 11T^{2} \)
13 \( 1 - 1.73T + 13T^{2} \)
17 \( 1 - 7.47T + 17T^{2} \)
19 \( 1 + 7.14T + 19T^{2} \)
29 \( 1 + 5.24T + 29T^{2} \)
31 \( 1 - 3.25T + 31T^{2} \)
37 \( 1 + 6.14T + 37T^{2} \)
41 \( 1 + 9.37T + 41T^{2} \)
43 \( 1 - 7.90T + 43T^{2} \)
47 \( 1 + 0.303T + 47T^{2} \)
53 \( 1 - 9.05T + 53T^{2} \)
59 \( 1 + 1.29T + 59T^{2} \)
61 \( 1 + 5.05T + 61T^{2} \)
67 \( 1 + 5.76T + 67T^{2} \)
71 \( 1 + 5.45T + 71T^{2} \)
73 \( 1 - 5.77T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 - 2.27T + 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−7.42845857203409253882636865833, −6.76349806131890918979818531445, −6.19872667858832642954744245581, −5.91760775215502960558423891060, −5.27274442031118899920902554436, −4.02216099625537690091580684816, −3.21664025566845862747079681106, −1.97021287394544814212500751474, −0.992074706346339689910436579230, 0, 0.992074706346339689910436579230, 1.97021287394544814212500751474, 3.21664025566845862747079681106, 4.02216099625537690091580684816, 5.27274442031118899920902554436, 5.91760775215502960558423891060, 6.19872667858832642954744245581, 6.76349806131890918979818531445, 7.42845857203409253882636865833

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