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 − 1.44·3-s + 4-s − 2.06·5-s + 1.44·6-s − 7-s − 8-s − 0.922·9-s + 2.06·10-s + 5.36·11-s − 1.44·12-s − 1.30·13-s + 14-s + 2.96·15-s + 16-s + 0.952·17-s + 0.922·18-s + 0.858·19-s − 2.06·20-s + 1.44·21-s − 5.36·22-s − 3.31·23-s + 1.44·24-s − 0.753·25-s + 1.30·26-s + 5.65·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.832·3-s + 0.5·4-s − 0.921·5-s + 0.588·6-s − 0.377·7-s − 0.353·8-s − 0.307·9-s + 0.651·10-s + 1.61·11-s − 0.416·12-s − 0.363·13-s + 0.267·14-s + 0.766·15-s + 0.250·16-s + 0.231·17-s + 0.217·18-s + 0.196·19-s − 0.460·20-s + 0.314·21-s − 1.14·22-s − 0.690·23-s + 0.294·24-s − 0.150·25-s + 0.256·26-s + 1.08·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5023726272$
$L(\frac12)$  $\approx$  $0.5023726272$
$L(\frac{3}{2})$   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 \( 1 + 1.44T + 3T^{2} \)
5 \( 1 + 2.06T + 5T^{2} \)
11 \( 1 - 5.36T + 11T^{2} \)
13 \( 1 + 1.30T + 13T^{2} \)
17 \( 1 - 0.952T + 17T^{2} \)
19 \( 1 - 0.858T + 19T^{2} \)
23 \( 1 + 3.31T + 23T^{2} \)
29 \( 1 + 6.46T + 29T^{2} \)
31 \( 1 + 1.34T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + 0.731T + 47T^{2} \)
53 \( 1 + 5.33T + 53T^{2} \)
59 \( 1 - 3.54T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 - 5.59T + 67T^{2} \)
71 \( 1 + 6.54T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 - 7.82T + 79T^{2} \)
83 \( 1 + 8.16T + 83T^{2} \)
89 \( 1 + 9.24T + 89T^{2} \)
97 \( 1 + 7.78T + 97T^{2} \)
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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

−8.050670626202152235844600066325, −7.38730413321485360470845871672, −6.68263062062088929718957824848, −6.09347050605605655036263150578, −5.43386107943305153817484484421, −4.27991698324218312230223240351, −3.75123824954431928918574817142, −2.75950320515702579646298228365, −1.49803644623818719168713976939, −0.44961642214952977618446974730, 0.44961642214952977618446974730, 1.49803644623818719168713976939, 2.75950320515702579646298228365, 3.75123824954431928918574817142, 4.27991698324218312230223240351, 5.43386107943305153817484484421, 6.09347050605605655036263150578, 6.68263062062088929718957824848, 7.38730413321485360470845871672, 8.050670626202152235844600066325

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