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 + 0.178·3-s + 4-s − 5-s + 0.178·6-s − 4.63·7-s + 8-s − 2.96·9-s − 10-s + 5.33·11-s + 0.178·12-s − 13-s − 4.63·14-s − 0.178·15-s + 16-s − 6.03·17-s − 2.96·18-s − 0.663·19-s − 20-s − 0.828·21-s + 5.33·22-s − 1.20·23-s + 0.178·24-s + 25-s − 26-s − 1.06·27-s − 4.63·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.103·3-s + 0.5·4-s − 0.447·5-s + 0.0730·6-s − 1.75·7-s + 0.353·8-s − 0.989·9-s − 0.316·10-s + 1.60·11-s + 0.0516·12-s − 0.277·13-s − 1.23·14-s − 0.0461·15-s + 0.250·16-s − 1.46·17-s − 0.699·18-s − 0.152·19-s − 0.223·20-s − 0.180·21-s + 1.13·22-s − 0.251·23-s + 0.0365·24-s + 0.200·25-s − 0.196·26-s − 0.205·27-s − 0.875·28-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  =  $(2,\ 4030,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.951111052$
$L(\frac12)$  $\approx$  $1.951111052$
$L(\frac{3}{2})$   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 \( 1 - 0.178T + 3T^{2} \)
7 \( 1 + 4.63T + 7T^{2} \)
11 \( 1 - 5.33T + 11T^{2} \)
17 \( 1 + 6.03T + 17T^{2} \)
19 \( 1 + 0.663T + 19T^{2} \)
23 \( 1 + 1.20T + 23T^{2} \)
29 \( 1 - 7.03T + 29T^{2} \)
37 \( 1 - 6.84T + 37T^{2} \)
41 \( 1 - 7.35T + 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 - 0.326T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 3.23T + 67T^{2} \)
71 \( 1 + 9.94T + 71T^{2} \)
73 \( 1 - 1.10T + 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 + 8.93T + 83T^{2} \)
89 \( 1 + 0.384T + 89T^{2} \)
97 \( 1 + 17.7T + 97T^{2} \)
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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

−8.627837792188475149174222112801, −7.47216834940710000320720190688, −6.69446324867810952824202706558, −6.30697354892488769666158151920, −5.67758270170868887737788607922, −4.26863760390875415386742308387, −4.02553634262998420332296781101, −2.97556825462200228714434827616, −2.43618260399829584679897347904, −0.68526156478619038567400605932, 0.68526156478619038567400605932, 2.43618260399829584679897347904, 2.97556825462200228714434827616, 4.02553634262998420332296781101, 4.26863760390875415386742308387, 5.67758270170868887737788607922, 6.30697354892488769666158151920, 6.69446324867810952824202706558, 7.47216834940710000320720190688, 8.627837792188475149174222112801

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