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 − 0.0412·3-s + 4-s − 3.47·5-s + 0.0412·6-s − 4.67·7-s − 8-s − 2.99·9-s + 3.47·10-s − 0.864·11-s − 0.0412·12-s + 1.77·13-s + 4.67·14-s + 0.143·15-s + 16-s + 0.0423·17-s + 2.99·18-s − 0.190·19-s − 3.47·20-s + 0.192·21-s + 0.864·22-s + 23-s + 0.0412·24-s + 7.07·25-s − 1.77·26-s + 0.247·27-s − 4.67·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0238·3-s + 0.5·4-s − 1.55·5-s + 0.0168·6-s − 1.76·7-s − 0.353·8-s − 0.999·9-s + 1.09·10-s − 0.260·11-s − 0.0119·12-s + 0.492·13-s + 1.24·14-s + 0.0370·15-s + 0.250·16-s + 0.0102·17-s + 0.706·18-s − 0.0436·19-s − 0.777·20-s + 0.0421·21-s + 0.184·22-s + 0.208·23-s + 0.00842·24-s + 1.41·25-s − 0.348·26-s + 0.0476·27-s − 0.883·28-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 \( 1 + 0.0412T + 3T^{2} \)
5 \( 1 + 3.47T + 5T^{2} \)
7 \( 1 + 4.67T + 7T^{2} \)
11 \( 1 + 0.864T + 11T^{2} \)
13 \( 1 - 1.77T + 13T^{2} \)
17 \( 1 - 0.0423T + 17T^{2} \)
19 \( 1 + 0.190T + 19T^{2} \)
29 \( 1 - 8.29T + 29T^{2} \)
31 \( 1 + 5.15T + 31T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 - 5.50T + 41T^{2} \)
43 \( 1 - 1.97T + 43T^{2} \)
47 \( 1 + 6.19T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 + 1.83T + 61T^{2} \)
67 \( 1 + 8.37T + 67T^{2} \)
71 \( 1 + 8.15T + 71T^{2} \)
73 \( 1 + 1.53T + 73T^{2} \)
79 \( 1 - 7.35T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + 3.01T + 89T^{2} \)
97 \( 1 - 9.58T + 97T^{2} \)
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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

−7.71256137229707919695241888152, −7.21176664397887693392521466158, −6.34282158713974621954285757723, −5.94309236397932775026763226454, −4.74029654923109786283864146581, −3.72935532835297955201610971288, −3.21314869125655492576321326997, −2.57533379243085219823600930185, −0.78074516153620293350883922369, 0, 0.78074516153620293350883922369, 2.57533379243085219823600930185, 3.21314869125655492576321326997, 3.72935532835297955201610971288, 4.74029654923109786283864146581, 5.94309236397932775026763226454, 6.34282158713974621954285757723, 7.21176664397887693392521466158, 7.71256137229707919695241888152

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