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 − 3.27·3-s + 4-s − 0.103·5-s + 3.27·6-s + 0.553·7-s − 8-s + 7.69·9-s + 0.103·10-s + 0.813·11-s − 3.27·12-s + 6.58·13-s − 0.553·14-s + 0.339·15-s + 16-s + 5.83·17-s − 7.69·18-s + 3.59·19-s − 0.103·20-s − 1.80·21-s − 0.813·22-s − 23-s + 3.27·24-s − 4.98·25-s − 6.58·26-s − 15.3·27-s + 0.553·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.88·3-s + 0.5·4-s − 0.0464·5-s + 1.33·6-s + 0.209·7-s − 0.353·8-s + 2.56·9-s + 0.0328·10-s + 0.245·11-s − 0.944·12-s + 1.82·13-s − 0.147·14-s + 0.0876·15-s + 0.250·16-s + 1.41·17-s − 1.81·18-s + 0.825·19-s − 0.0232·20-s − 0.394·21-s − 0.173·22-s − 0.208·23-s + 0.667·24-s − 0.997·25-s − 1.29·26-s − 2.95·27-s + 0.104·28-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 \( 1 + 3.27T + 3T^{2} \)
5 \( 1 + 0.103T + 5T^{2} \)
7 \( 1 - 0.553T + 7T^{2} \)
11 \( 1 - 0.813T + 11T^{2} \)
13 \( 1 - 6.58T + 13T^{2} \)
17 \( 1 - 5.83T + 17T^{2} \)
19 \( 1 - 3.59T + 19T^{2} \)
29 \( 1 + 5.37T + 29T^{2} \)
31 \( 1 + 7.85T + 31T^{2} \)
37 \( 1 - 0.849T + 37T^{2} \)
41 \( 1 - 2.88T + 41T^{2} \)
43 \( 1 + 6.56T + 43T^{2} \)
47 \( 1 + 4.80T + 47T^{2} \)
53 \( 1 - 3.40T + 53T^{2} \)
59 \( 1 + 3.93T + 59T^{2} \)
61 \( 1 - 3.68T + 61T^{2} \)
67 \( 1 - 0.873T + 67T^{2} \)
71 \( 1 - 6.26T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 - 5.08T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + 1.29T + 89T^{2} \)
97 \( 1 + 5.53T + 97T^{2} \)
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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

−7.63501998189065409227871433363, −6.97196376782227847286440504862, −6.19024401006913602771079921763, −5.68570587523050565732048130055, −5.24538558437079832441686360071, −4.02063789011396172232568991157, −3.46367226733141550543558406357, −1.60868488964204811591381092580, −1.20713432643807350331306966009, 0, 1.20713432643807350331306966009, 1.60868488964204811591381092580, 3.46367226733141550543558406357, 4.02063789011396172232568991157, 5.24538558437079832441686360071, 5.68570587523050565732048130055, 6.19024401006913602771079921763, 6.97196376782227847286440504862, 7.63501998189065409227871433363

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