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 − 2.67·3-s + 4-s + 3.15·5-s + 2.67·6-s − 7-s − 8-s + 4.13·9-s − 3.15·10-s + 3.01·11-s − 2.67·12-s − 4.63·13-s + 14-s − 8.42·15-s + 16-s − 6.60·17-s − 4.13·18-s − 4.22·19-s + 3.15·20-s + 2.67·21-s − 3.01·22-s − 0.524·23-s + 2.67·24-s + 4.96·25-s + 4.63·26-s − 3.02·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.54·3-s + 0.5·4-s + 1.41·5-s + 1.09·6-s − 0.377·7-s − 0.353·8-s + 1.37·9-s − 0.998·10-s + 0.909·11-s − 0.770·12-s − 1.28·13-s + 0.267·14-s − 2.17·15-s + 0.250·16-s − 1.60·17-s − 0.973·18-s − 0.968·19-s + 0.705·20-s + 0.582·21-s − 0.643·22-s − 0.109·23-s + 0.545·24-s + 0.992·25-s + 0.908·26-s − 0.581·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5314497081$
$L(\frac12)$  $\approx$  $0.5314497081$
$L(\frac{3}{2})$   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 \( 1 + 2.67T + 3T^{2} \)
5 \( 1 - 3.15T + 5T^{2} \)
11 \( 1 - 3.01T + 11T^{2} \)
13 \( 1 + 4.63T + 13T^{2} \)
17 \( 1 + 6.60T + 17T^{2} \)
19 \( 1 + 4.22T + 19T^{2} \)
23 \( 1 + 0.524T + 23T^{2} \)
29 \( 1 + 9.21T + 29T^{2} \)
31 \( 1 - 1.00T + 31T^{2} \)
37 \( 1 + 8.56T + 37T^{2} \)
41 \( 1 - 6.31T + 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + 7.89T + 47T^{2} \)
53 \( 1 - 0.831T + 53T^{2} \)
59 \( 1 + 2.18T + 59T^{2} \)
61 \( 1 + 1.29T + 61T^{2} \)
67 \( 1 - 8.28T + 67T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 - 16.6T + 73T^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 - 7.07T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−8.085138796048395922881934917959, −6.89485565348888102261259107800, −6.62574874682223411660208661701, −6.20746643519936200161832624362, −5.29197013459696970911197311934, −4.84387658416904560491655533398, −3.71105706278387183756120201148, −2.19309079660110394257686862049, −1.84606255616711857496106454765, −0.44342655580290455017389707574, 0.44342655580290455017389707574, 1.84606255616711857496106454765, 2.19309079660110394257686862049, 3.71105706278387183756120201148, 4.84387658416904560491655533398, 5.29197013459696970911197311934, 6.20746643519936200161832624362, 6.62574874682223411660208661701, 6.89485565348888102261259107800, 8.085138796048395922881934917959

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