Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 431 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3.32·3-s + 4-s − 1.53·5-s − 3.32·6-s − 7-s − 8-s + 8.07·9-s + 1.53·10-s − 0.220·11-s + 3.32·12-s − 0.413·13-s + 14-s − 5.11·15-s + 16-s − 4.27·17-s − 8.07·18-s + 0.560·19-s − 1.53·20-s − 3.32·21-s + 0.220·22-s + 6.61·23-s − 3.32·24-s − 2.63·25-s + 0.413·26-s + 16.8·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.92·3-s + 0.5·4-s − 0.687·5-s − 1.35·6-s − 0.377·7-s − 0.353·8-s + 2.69·9-s + 0.486·10-s − 0.0664·11-s + 0.960·12-s − 0.114·13-s + 0.267·14-s − 1.32·15-s + 0.250·16-s − 1.03·17-s − 1.90·18-s + 0.128·19-s − 0.343·20-s − 0.726·21-s + 0.0469·22-s + 1.37·23-s − 0.679·24-s − 0.527·25-s + 0.0810·26-s + 3.24·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6034\)    =    \(2 \cdot 7 \cdot 431\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.572439104$
$L(\frac12)$  $\approx$  $2.572439104$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;431\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;431\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
7 \( 1 + T \)
431 \( 1 - T \)
good3 \( 1 - 3.32T + 3T^{2} \)
5 \( 1 + 1.53T + 5T^{2} \)
11 \( 1 + 0.220T + 11T^{2} \)
13 \( 1 + 0.413T + 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
19 \( 1 - 0.560T + 19T^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 + 3.42T + 29T^{2} \)
31 \( 1 - 6.99T + 31T^{2} \)
37 \( 1 + 6.87T + 37T^{2} \)
41 \( 1 - 8.27T + 41T^{2} \)
43 \( 1 + 0.779T + 43T^{2} \)
47 \( 1 - 5.40T + 47T^{2} \)
53 \( 1 - 8.45T + 53T^{2} \)
59 \( 1 + 3.80T + 59T^{2} \)
61 \( 1 - 3.36T + 61T^{2} \)
67 \( 1 - 8.00T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 - 1.20T + 73T^{2} \)
79 \( 1 - 0.0748T + 79T^{2} \)
83 \( 1 + 6.38T + 83T^{2} \)
89 \( 1 - 3.92T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−8.253656939543277148078167878935, −7.44502655629872052468616451286, −7.15557339693177507111142662018, −6.35147574279672401181356727036, −4.97251568433581729101809147547, −4.06745092014687789964121195729, −3.49491998937107372579874796118, −2.66690130212002233136163626244, −2.08217549301289261417115868223, −0.852565163474016829081619831074, 0.852565163474016829081619831074, 2.08217549301289261417115868223, 2.66690130212002233136163626244, 3.49491998937107372579874796118, 4.06745092014687789964121195729, 4.97251568433581729101809147547, 6.35147574279672401181356727036, 7.15557339693177507111142662018, 7.44502655629872052468616451286, 8.253656939543277148078167878935

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