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 − 2.98·3-s + 4-s + 1.57·5-s + 2.98·6-s + 2.90·7-s − 8-s + 5.90·9-s − 1.57·10-s − 4.95·11-s − 2.98·12-s − 4.16·13-s − 2.90·14-s − 4.71·15-s + 16-s + 2.92·17-s − 5.90·18-s + 2.68·19-s + 1.57·20-s − 8.68·21-s + 4.95·22-s + 23-s + 2.98·24-s − 2.50·25-s + 4.16·26-s − 8.68·27-s + 2.90·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.72·3-s + 0.5·4-s + 0.706·5-s + 1.21·6-s + 1.09·7-s − 0.353·8-s + 1.96·9-s − 0.499·10-s − 1.49·11-s − 0.861·12-s − 1.15·13-s − 0.777·14-s − 1.21·15-s + 0.250·16-s + 0.708·17-s − 1.39·18-s + 0.616·19-s + 0.353·20-s − 1.89·21-s + 1.05·22-s + 0.208·23-s + 0.609·24-s − 0.500·25-s + 0.816·26-s − 1.67·27-s + 0.549·28-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 \( 1 + 2.98T + 3T^{2} \)
5 \( 1 - 1.57T + 5T^{2} \)
7 \( 1 - 2.90T + 7T^{2} \)
11 \( 1 + 4.95T + 11T^{2} \)
13 \( 1 + 4.16T + 13T^{2} \)
17 \( 1 - 2.92T + 17T^{2} \)
19 \( 1 - 2.68T + 19T^{2} \)
29 \( 1 - 5.30T + 29T^{2} \)
31 \( 1 + 3.13T + 31T^{2} \)
37 \( 1 - 4.98T + 37T^{2} \)
41 \( 1 - 6.04T + 41T^{2} \)
43 \( 1 + 0.659T + 43T^{2} \)
47 \( 1 + 6.46T + 47T^{2} \)
53 \( 1 - 1.53T + 53T^{2} \)
59 \( 1 + 3.36T + 59T^{2} \)
61 \( 1 + 8.26T + 61T^{2} \)
67 \( 1 + 4.81T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 - 1.36T + 73T^{2} \)
79 \( 1 + 1.09T + 79T^{2} \)
83 \( 1 + 5.31T + 83T^{2} \)
89 \( 1 - 9.76T + 89T^{2} \)
97 \( 1 - 0.168T + 97T^{2} \)
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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

−7.58772308965677213708321259259, −7.19254049659460043217611834204, −6.07802577221460491766900404505, −5.67565772388508867473710112350, −4.97302316830835707472342486741, −4.60639293603982442081447994520, −2.92474103780466834284311634814, −1.97839535149083486985268143653, −1.09255648901912492308391020268, 0, 1.09255648901912492308391020268, 1.97839535149083486985268143653, 2.92474103780466834284311634814, 4.60639293603982442081447994520, 4.97302316830835707472342486741, 5.67565772388508867473710112350, 6.07802577221460491766900404505, 7.19254049659460043217611834204, 7.58772308965677213708321259259

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