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 + 2.22·3-s + 4-s − 3.05·5-s − 2.22·6-s + 0.802·7-s − 8-s + 1.96·9-s + 3.05·10-s + 3.73·11-s + 2.22·12-s − 0.852·13-s − 0.802·14-s − 6.80·15-s + 16-s − 4.27·17-s − 1.96·18-s + 2.53·19-s − 3.05·20-s + 1.78·21-s − 3.73·22-s + 23-s − 2.22·24-s + 4.34·25-s + 0.852·26-s − 2.30·27-s + 0.802·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.28·3-s + 0.5·4-s − 1.36·5-s − 0.909·6-s + 0.303·7-s − 0.353·8-s + 0.654·9-s + 0.966·10-s + 1.12·11-s + 0.643·12-s − 0.236·13-s − 0.214·14-s − 1.75·15-s + 0.250·16-s − 1.03·17-s − 0.462·18-s + 0.580·19-s − 0.683·20-s + 0.389·21-s − 0.795·22-s + 0.208·23-s − 0.454·24-s + 0.868·25-s + 0.167·26-s − 0.444·27-s + 0.151·28-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 \( 1 - 2.22T + 3T^{2} \)
5 \( 1 + 3.05T + 5T^{2} \)
7 \( 1 - 0.802T + 7T^{2} \)
11 \( 1 - 3.73T + 11T^{2} \)
13 \( 1 + 0.852T + 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
19 \( 1 - 2.53T + 19T^{2} \)
29 \( 1 + 0.484T + 29T^{2} \)
31 \( 1 + 8.25T + 31T^{2} \)
37 \( 1 - 1.59T + 37T^{2} \)
41 \( 1 - 1.30T + 41T^{2} \)
43 \( 1 + 6.86T + 43T^{2} \)
47 \( 1 - 0.570T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 - 8.46T + 59T^{2} \)
61 \( 1 + 0.408T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 7.59T + 73T^{2} \)
79 \( 1 + 5.22T + 79T^{2} \)
83 \( 1 - 3.11T + 83T^{2} \)
89 \( 1 + 6.56T + 89T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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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

−7.953620596746301671179731184571, −7.16912521839925297415406719687, −6.86145959809748533491354732486, −5.63098988471750948884905989288, −4.47681354846891425539593722693, −3.83592686809651904244294910832, −3.25028673996343814860746452296, −2.30608921857918143949298785034, −1.37823128279622864792447388545, 0, 1.37823128279622864792447388545, 2.30608921857918143949298785034, 3.25028673996343814860746452296, 3.83592686809651904244294910832, 4.47681354846891425539593722693, 5.63098988471750948884905989288, 6.86145959809748533491354732486, 7.16912521839925297415406719687, 7.953620596746301671179731184571

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