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 + 1.76·3-s + 4-s − 0.379·5-s − 1.76·6-s − 0.00141·7-s − 8-s + 0.121·9-s + 0.379·10-s − 1.92·11-s + 1.76·12-s − 0.508·13-s + 0.00141·14-s − 0.670·15-s + 16-s − 3.60·17-s − 0.121·18-s + 5.47·19-s − 0.379·20-s − 0.00249·21-s + 1.92·22-s + 23-s − 1.76·24-s − 4.85·25-s + 0.508·26-s − 5.08·27-s − 0.00141·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.02·3-s + 0.5·4-s − 0.169·5-s − 0.721·6-s − 0.000534·7-s − 0.353·8-s + 0.0406·9-s + 0.119·10-s − 0.580·11-s + 0.510·12-s − 0.141·13-s + 0.000378·14-s − 0.173·15-s + 0.250·16-s − 0.874·17-s − 0.0287·18-s + 1.25·19-s − 0.0848·20-s − 0.000545·21-s + 0.410·22-s + 0.208·23-s − 0.360·24-s − 0.971·25-s + 0.0997·26-s − 0.978·27-s − 0.000267·28-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 \( 1 - 1.76T + 3T^{2} \)
5 \( 1 + 0.379T + 5T^{2} \)
7 \( 1 + 0.00141T + 7T^{2} \)
11 \( 1 + 1.92T + 11T^{2} \)
13 \( 1 + 0.508T + 13T^{2} \)
17 \( 1 + 3.60T + 17T^{2} \)
19 \( 1 - 5.47T + 19T^{2} \)
29 \( 1 - 1.18T + 29T^{2} \)
31 \( 1 - 7.55T + 31T^{2} \)
37 \( 1 + 0.946T + 37T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 + 2.24T + 43T^{2} \)
47 \( 1 - 0.668T + 47T^{2} \)
53 \( 1 - 0.305T + 53T^{2} \)
59 \( 1 - 8.06T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 - 4.52T + 67T^{2} \)
71 \( 1 + 2.54T + 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 + 5.09T + 89T^{2} \)
97 \( 1 + 18.6T + 97T^{2} \)
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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

−7.74366150365599769559506607934, −7.43923113787400139159111624984, −6.46632989069912072069076894401, −5.69270953401072059964088097958, −4.77143776834611868883627174195, −3.83074588575718909782312122515, −2.91172990356652271556074942044, −2.47504701496631892775441663118, −1.37670636111020642304279110276, 0, 1.37670636111020642304279110276, 2.47504701496631892775441663118, 2.91172990356652271556074942044, 3.83074588575718909782312122515, 4.77143776834611868883627174195, 5.69270953401072059964088097958, 6.46632989069912072069076894401, 7.43923113787400139159111624984, 7.74366150365599769559506607934

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