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 + 0.430·3-s + 4-s + 1.34·5-s − 0.430·6-s − 7-s − 8-s − 2.81·9-s − 1.34·10-s + 4.88·11-s + 0.430·12-s + 4.15·13-s + 14-s + 0.577·15-s + 16-s + 4.03·17-s + 2.81·18-s − 0.649·19-s + 1.34·20-s − 0.430·21-s − 4.88·22-s − 2.19·23-s − 0.430·24-s − 3.20·25-s − 4.15·26-s − 2.50·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.248·3-s + 0.5·4-s + 0.599·5-s − 0.175·6-s − 0.377·7-s − 0.353·8-s − 0.938·9-s − 0.423·10-s + 1.47·11-s + 0.124·12-s + 1.15·13-s + 0.267·14-s + 0.149·15-s + 0.250·16-s + 0.979·17-s + 0.663·18-s − 0.149·19-s + 0.299·20-s − 0.0940·21-s − 1.04·22-s − 0.458·23-s − 0.0879·24-s − 0.640·25-s − 0.815·26-s − 0.482·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.866600609$
$L(\frac12)$  $\approx$  $1.866600609$
$L(\frac{3}{2})$   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 \( 1 - 0.430T + 3T^{2} \)
5 \( 1 - 1.34T + 5T^{2} \)
11 \( 1 - 4.88T + 11T^{2} \)
13 \( 1 - 4.15T + 13T^{2} \)
17 \( 1 - 4.03T + 17T^{2} \)
19 \( 1 + 0.649T + 19T^{2} \)
23 \( 1 + 2.19T + 23T^{2} \)
29 \( 1 + 0.741T + 29T^{2} \)
31 \( 1 + 0.702T + 31T^{2} \)
37 \( 1 - 5.70T + 37T^{2} \)
41 \( 1 - 7.65T + 41T^{2} \)
43 \( 1 - 9.49T + 43T^{2} \)
47 \( 1 - 6.32T + 47T^{2} \)
53 \( 1 + 9.84T + 53T^{2} \)
59 \( 1 + 0.936T + 59T^{2} \)
61 \( 1 + 7.74T + 61T^{2} \)
67 \( 1 + 6.50T + 67T^{2} \)
71 \( 1 + 0.0975T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + 17.7T + 97T^{2} \)
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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

−8.103166612580033528788019314569, −7.59311791950596111905378961424, −6.51192282587690986406644438544, −6.02738088181478887621628577432, −5.66472881885965742131490624469, −4.19144641826844715609461597900, −3.53711832634225580391879081816, −2.68289093097309813103158581088, −1.70506449092377827771532945809, −0.824166769725427444514092361864, 0.824166769725427444514092361864, 1.70506449092377827771532945809, 2.68289093097309813103158581088, 3.53711832634225580391879081816, 4.19144641826844715609461597900, 5.66472881885965742131490624469, 6.02738088181478887621628577432, 6.51192282587690986406644438544, 7.59311791950596111905378961424, 8.103166612580033528788019314569

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