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 − 2.34·3-s + 4-s − 0.801·5-s + 2.34·6-s + 2.16·7-s − 8-s + 2.50·9-s + 0.801·10-s + 4.95·11-s − 2.34·12-s − 1.05·13-s − 2.16·14-s + 1.88·15-s + 16-s − 2.72·17-s − 2.50·18-s + 5.72·19-s − 0.801·20-s − 5.09·21-s − 4.95·22-s + 23-s + 2.34·24-s − 4.35·25-s + 1.05·26-s + 1.15·27-s + 2.16·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.35·3-s + 0.5·4-s − 0.358·5-s + 0.957·6-s + 0.819·7-s − 0.353·8-s + 0.835·9-s + 0.253·10-s + 1.49·11-s − 0.677·12-s − 0.292·13-s − 0.579·14-s + 0.485·15-s + 0.250·16-s − 0.660·17-s − 0.590·18-s + 1.31·19-s − 0.179·20-s − 1.11·21-s − 1.05·22-s + 0.208·23-s + 0.478·24-s − 0.871·25-s + 0.206·26-s + 0.223·27-s + 0.409·28-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 \( 1 + 2.34T + 3T^{2} \)
5 \( 1 + 0.801T + 5T^{2} \)
7 \( 1 - 2.16T + 7T^{2} \)
11 \( 1 - 4.95T + 11T^{2} \)
13 \( 1 + 1.05T + 13T^{2} \)
17 \( 1 + 2.72T + 17T^{2} \)
19 \( 1 - 5.72T + 19T^{2} \)
29 \( 1 + 3.63T + 29T^{2} \)
31 \( 1 + 1.85T + 31T^{2} \)
37 \( 1 - 1.54T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 3.49T + 43T^{2} \)
47 \( 1 - 9.13T + 47T^{2} \)
53 \( 1 - 5.24T + 53T^{2} \)
59 \( 1 + 0.391T + 59T^{2} \)
61 \( 1 + 8.80T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + 6.56T + 71T^{2} \)
73 \( 1 + 3.25T + 73T^{2} \)
79 \( 1 + 4.15T + 79T^{2} \)
83 \( 1 + 1.60T + 83T^{2} \)
89 \( 1 + 3.89T + 89T^{2} \)
97 \( 1 + 9.72T + 97T^{2} \)
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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

−7.45916155561052571154016450979, −7.18075148455282361306240466225, −6.29475903943038356119318818963, −5.72303480065223055295746557191, −4.92521069837461316528445890293, −4.22380192276196106996497967903, −3.26965886291067570421233513198, −1.84197387446626724699596144305, −1.13429550678216000632682495509, 0, 1.13429550678216000632682495509, 1.84197387446626724699596144305, 3.26965886291067570421233513198, 4.22380192276196106996497967903, 4.92521069837461316528445890293, 5.72303480065223055295746557191, 6.29475903943038356119318818963, 7.18075148455282361306240466225, 7.45916155561052571154016450979

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