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 + 1.56·3-s + 4-s − 1.28·5-s − 1.56·6-s − 7-s − 8-s − 0.563·9-s + 1.28·10-s − 0.685·11-s + 1.56·12-s − 1.48·13-s + 14-s − 2.00·15-s + 16-s − 5.29·17-s + 0.563·18-s − 3.49·19-s − 1.28·20-s − 1.56·21-s + 0.685·22-s + 4.10·23-s − 1.56·24-s − 3.34·25-s + 1.48·26-s − 5.56·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.901·3-s + 0.5·4-s − 0.575·5-s − 0.637·6-s − 0.377·7-s − 0.353·8-s − 0.187·9-s + 0.407·10-s − 0.206·11-s + 0.450·12-s − 0.411·13-s + 0.267·14-s − 0.518·15-s + 0.250·16-s − 1.28·17-s + 0.132·18-s − 0.802·19-s − 0.287·20-s − 0.340·21-s + 0.146·22-s + 0.856·23-s − 0.318·24-s − 0.668·25-s + 0.290·26-s − 1.07·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9603886715$
$L(\frac12)$  $\approx$  $0.9603886715$
$L(\frac{3}{2})$   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 \( 1 - 1.56T + 3T^{2} \)
5 \( 1 + 1.28T + 5T^{2} \)
11 \( 1 + 0.685T + 11T^{2} \)
13 \( 1 + 1.48T + 13T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 + 3.49T + 19T^{2} \)
23 \( 1 - 4.10T + 23T^{2} \)
29 \( 1 - 0.0567T + 29T^{2} \)
31 \( 1 + 6.65T + 31T^{2} \)
37 \( 1 - 2.52T + 37T^{2} \)
41 \( 1 + 1.28T + 41T^{2} \)
43 \( 1 - 9.13T + 43T^{2} \)
47 \( 1 - 3.53T + 47T^{2} \)
53 \( 1 - 6.42T + 53T^{2} \)
59 \( 1 - 7.80T + 59T^{2} \)
61 \( 1 - 5.85T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + 6.53T + 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 - 1.36T + 89T^{2} \)
97 \( 1 + 6.23T + 97T^{2} \)
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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

−8.204479867201185958556686939151, −7.47944128939484266678311801729, −6.97651937368084258904121373744, −6.13128572956440713938825486824, −5.25149971697968853419124606724, −4.14878412328748728670215207651, −3.56691079841657386565999514315, −2.52156326911436302926813179334, −2.13341776086793481722700618568, −0.51152000154552941138218756977, 0.51152000154552941138218756977, 2.13341776086793481722700618568, 2.52156326911436302926813179334, 3.56691079841657386565999514315, 4.14878412328748728670215207651, 5.25149971697968853419124606724, 6.13128572956440713938825486824, 6.97651937368084258904121373744, 7.47944128939484266678311801729, 8.204479867201185958556686939151

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