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 − 1.91·3-s + 4-s + 0.374·5-s + 1.91·6-s − 7-s − 8-s + 0.678·9-s − 0.374·10-s − 1.13·11-s − 1.91·12-s + 0.0849·13-s + 14-s − 0.718·15-s + 16-s − 4.64·17-s − 0.678·18-s + 4.04·19-s + 0.374·20-s + 1.91·21-s + 1.13·22-s + 0.619·23-s + 1.91·24-s − 4.85·25-s − 0.0849·26-s + 4.45·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.10·3-s + 0.5·4-s + 0.167·5-s + 0.782·6-s − 0.377·7-s − 0.353·8-s + 0.226·9-s − 0.118·10-s − 0.341·11-s − 0.553·12-s + 0.0235·13-s + 0.267·14-s − 0.185·15-s + 0.250·16-s − 1.12·17-s − 0.159·18-s + 0.928·19-s + 0.0837·20-s + 0.418·21-s + 0.241·22-s + 0.129·23-s + 0.391·24-s − 0.971·25-s − 0.0166·26-s + 0.856·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4674987109$
$L(\frac12)$  $\approx$  $0.4674987109$
$L(\frac{3}{2})$   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 \( 1 + 1.91T + 3T^{2} \)
5 \( 1 - 0.374T + 5T^{2} \)
11 \( 1 + 1.13T + 11T^{2} \)
13 \( 1 - 0.0849T + 13T^{2} \)
17 \( 1 + 4.64T + 17T^{2} \)
19 \( 1 - 4.04T + 19T^{2} \)
23 \( 1 - 0.619T + 23T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 + 2.63T + 31T^{2} \)
37 \( 1 + 5.53T + 37T^{2} \)
41 \( 1 - 0.897T + 41T^{2} \)
43 \( 1 + 3.73T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 + 9.09T + 53T^{2} \)
59 \( 1 - 1.43T + 59T^{2} \)
61 \( 1 - 7.69T + 61T^{2} \)
67 \( 1 + 4.04T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 - 0.0269T + 73T^{2} \)
79 \( 1 + 3.39T + 79T^{2} \)
83 \( 1 - 0.337T + 83T^{2} \)
89 \( 1 + 3.97T + 89T^{2} \)
97 \( 1 + 5.17T + 97T^{2} \)
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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

−8.165372150599304032205147800936, −7.24746726176004014042144383755, −6.57734011078434308311801210619, −6.15279270935845511815531961683, −5.27507098411579282258403751235, −4.74405389287157467194864640768, −3.52670913270928080971540301140, −2.65282084056799748604045745364, −1.59058536422875320171964485840, −0.42354721186780244481109080566, 0.42354721186780244481109080566, 1.59058536422875320171964485840, 2.65282084056799748604045745364, 3.52670913270928080971540301140, 4.74405389287157467194864640768, 5.27507098411579282258403751235, 6.15279270935845511815531961683, 6.57734011078434308311801210619, 7.24746726176004014042144383755, 8.165372150599304032205147800936

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