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 + 3.05·3-s + 4-s − 5-s + 3.05·6-s + 4.02·7-s + 8-s + 6.32·9-s − 10-s − 0.362·11-s + 3.05·12-s − 13-s + 4.02·14-s − 3.05·15-s + 16-s − 1.92·17-s + 6.32·18-s − 4.56·19-s − 20-s + 12.2·21-s − 0.362·22-s + 1.98·23-s + 3.05·24-s + 25-s − 26-s + 10.1·27-s + 4.02·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.76·3-s + 0.5·4-s − 0.447·5-s + 1.24·6-s + 1.52·7-s + 0.353·8-s + 2.10·9-s − 0.316·10-s − 0.109·11-s + 0.881·12-s − 0.277·13-s + 1.07·14-s − 0.788·15-s + 0.250·16-s − 0.466·17-s + 1.49·18-s − 1.04·19-s − 0.223·20-s + 2.68·21-s − 0.0773·22-s + 0.414·23-s + 0.623·24-s + 0.200·25-s − 0.196·26-s + 1.95·27-s + 0.760·28-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  =  $(2,\ 4030,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.416643539$
$L(\frac12)$  $\approx$  $6.416643539$
$L(\frac{3}{2})$   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 \( 1 - 3.05T + 3T^{2} \)
7 \( 1 - 4.02T + 7T^{2} \)
11 \( 1 + 0.362T + 11T^{2} \)
17 \( 1 + 1.92T + 17T^{2} \)
19 \( 1 + 4.56T + 19T^{2} \)
23 \( 1 - 1.98T + 23T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 - 1.00T + 41T^{2} \)
43 \( 1 - 1.27T + 43T^{2} \)
47 \( 1 + 1.42T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 - 3.80T + 67T^{2} \)
71 \( 1 - 6.68T + 71T^{2} \)
73 \( 1 - 7.24T + 73T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 - 0.0901T + 83T^{2} \)
89 \( 1 - 1.30T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−8.301960768523917765443842021566, −7.87308310459632531731192836399, −7.20767385319245678422398246763, −6.38224514120485302499778908698, −5.03787082406875078976045599232, −4.49421288663994001608743668466, −3.91552488322049133849561065428, −2.89826261053644904956625560025, −2.25596595565586168057176863628, −1.40669157213978022784888625685, 1.40669157213978022784888625685, 2.25596595565586168057176863628, 2.89826261053644904956625560025, 3.91552488322049133849561065428, 4.49421288663994001608743668466, 5.03787082406875078976045599232, 6.38224514120485302499778908698, 7.20767385319245678422398246763, 7.87308310459632531731192836399, 8.301960768523917765443842021566

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