Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 \cdot 31 $
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 + 2.63·3-s + 4-s − 5-s + 2.63·6-s + 0.203·7-s + 8-s + 3.94·9-s − 10-s + 4.83·11-s + 2.63·12-s − 13-s + 0.203·14-s − 2.63·15-s + 16-s − 0.0683·17-s + 3.94·18-s + 3.00·19-s − 20-s + 0.535·21-s + 4.83·22-s + 4.98·23-s + 2.63·24-s + 25-s − 26-s + 2.48·27-s + 0.203·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.52·3-s + 0.5·4-s − 0.447·5-s + 1.07·6-s + 0.0767·7-s + 0.353·8-s + 1.31·9-s − 0.316·10-s + 1.45·11-s + 0.760·12-s − 0.277·13-s + 0.0542·14-s − 0.680·15-s + 0.250·16-s − 0.0165·17-s + 0.929·18-s + 0.689·19-s − 0.223·20-s + 0.116·21-s + 1.03·22-s + 1.03·23-s + 0.537·24-s + 0.200·25-s − 0.196·26-s + 0.479·27-s + 0.0383·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4030} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4030,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.537886014$
$L(\frac12)$  $\approx$  $5.537886014$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 + T \)
31 \( 1 + T \)
good3 \( 1 - 2.63T + 3T^{2} \)
7 \( 1 - 0.203T + 7T^{2} \)
11 \( 1 - 4.83T + 11T^{2} \)
17 \( 1 + 0.0683T + 17T^{2} \)
19 \( 1 - 3.00T + 19T^{2} \)
23 \( 1 - 4.98T + 23T^{2} \)
29 \( 1 + 4.67T + 29T^{2} \)
37 \( 1 - 6.73T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 + 0.517T + 43T^{2} \)
47 \( 1 - 2.49T + 47T^{2} \)
53 \( 1 - 7.40T + 53T^{2} \)
59 \( 1 + 9.25T + 59T^{2} \)
61 \( 1 - 8.56T + 61T^{2} \)
67 \( 1 + 5.12T + 67T^{2} \)
71 \( 1 - 1.20T + 71T^{2} \)
73 \( 1 + 2.64T + 73T^{2} \)
79 \( 1 - 7.88T + 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + 1.83T + 97T^{2} \)
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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

−8.399566772174344469502697066030, −7.68180104754358004561513356708, −7.08472374231593728423983499519, −6.40479159208358306258841269555, −5.26223946966435149461906038226, −4.41984412128042109993725296034, −3.63069521052672956646084222740, −3.23448033319172831544523815564, −2.21788518626850889625733199442, −1.26656015597652495730999352302, 1.26656015597652495730999352302, 2.21788518626850889625733199442, 3.23448033319172831544523815564, 3.63069521052672956646084222740, 4.41984412128042109993725296034, 5.26223946966435149461906038226, 6.40479159208358306258841269555, 7.08472374231593728423983499519, 7.68180104754358004561513356708, 8.399566772174344469502697066030

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