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 − 1.69·3-s + 4-s + 3.76·5-s + 1.69·6-s − 7-s − 8-s − 0.129·9-s − 3.76·10-s + 5.13·11-s − 1.69·12-s + 2.53·13-s + 14-s − 6.37·15-s + 16-s + 1.63·17-s + 0.129·18-s + 2.24·19-s + 3.76·20-s + 1.69·21-s − 5.13·22-s + 4.75·23-s + 1.69·24-s + 9.17·25-s − 2.53·26-s + 5.30·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.978·3-s + 0.5·4-s + 1.68·5-s + 0.691·6-s − 0.377·7-s − 0.353·8-s − 0.0432·9-s − 1.19·10-s + 1.54·11-s − 0.489·12-s + 0.702·13-s + 0.267·14-s − 1.64·15-s + 0.250·16-s + 0.396·17-s + 0.0305·18-s + 0.514·19-s + 0.841·20-s + 0.369·21-s − 1.09·22-s + 0.990·23-s + 0.345·24-s + 1.83·25-s − 0.496·26-s + 1.02·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.742832669$
$L(\frac12)$  $\approx$  $1.742832669$
$L(\frac{3}{2})$   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 \( 1 + 1.69T + 3T^{2} \)
5 \( 1 - 3.76T + 5T^{2} \)
11 \( 1 - 5.13T + 11T^{2} \)
13 \( 1 - 2.53T + 13T^{2} \)
17 \( 1 - 1.63T + 17T^{2} \)
19 \( 1 - 2.24T + 19T^{2} \)
23 \( 1 - 4.75T + 23T^{2} \)
29 \( 1 - 3.82T + 29T^{2} \)
31 \( 1 - 7.14T + 31T^{2} \)
37 \( 1 - 2.28T + 37T^{2} \)
41 \( 1 + 6.10T + 41T^{2} \)
43 \( 1 + 7.09T + 43T^{2} \)
47 \( 1 - 1.29T + 47T^{2} \)
53 \( 1 - 1.88T + 53T^{2} \)
59 \( 1 - 7.44T + 59T^{2} \)
61 \( 1 + 0.00800T + 61T^{2} \)
67 \( 1 - 0.701T + 67T^{2} \)
71 \( 1 + 3.28T + 71T^{2} \)
73 \( 1 + 2.75T + 73T^{2} \)
79 \( 1 - 3.70T + 79T^{2} \)
83 \( 1 - 7.74T + 83T^{2} \)
89 \( 1 + 8.90T + 89T^{2} \)
97 \( 1 + 8.92T + 97T^{2} \)
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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

−8.373272127351347084406694017250, −6.95588123168913940027752144987, −6.59129979760453737847868724744, −6.09470170625937521981661358392, −5.50617310832830917964166944607, −4.73377186279486115003617630146, −3.44530645884197015357206112001, −2.60877513025620116338970623092, −1.41193450770076352511436363706, −0.943225675075969223661102077749, 0.943225675075969223661102077749, 1.41193450770076352511436363706, 2.60877513025620116338970623092, 3.44530645884197015357206112001, 4.73377186279486115003617630146, 5.50617310832830917964166944607, 6.09470170625937521981661358392, 6.59129979760453737847868724744, 6.95588123168913940027752144987, 8.373272127351347084406694017250

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