Properties

Degree 2
Conductor $ 5 \cdot 13 \cdot 31 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.709·2-s − 1.77·3-s − 0.497·4-s + 5-s + 1.25·6-s + 0.241·7-s + 1.06·8-s + 2.13·9-s − 0.709·10-s + 1.94·11-s + 0.880·12-s − 13-s − 0.170·14-s − 1.77·15-s − 0.255·16-s + 1.49·17-s − 1.51·18-s − 0.497·20-s − 0.426·21-s − 1.37·22-s − 1.13·23-s − 1.88·24-s + 25-s + 0.709·26-s − 2.01·27-s − 0.119·28-s + 1.25·30-s + ⋯
L(s)  = 1  − 0.709·2-s − 1.77·3-s − 0.497·4-s + 5-s + 1.25·6-s + 0.241·7-s + 1.06·8-s + 2.13·9-s − 0.709·10-s + 1.94·11-s + 0.880·12-s − 13-s − 0.170·14-s − 1.77·15-s − 0.255·16-s + 1.49·17-s − 1.51·18-s − 0.497·20-s − 0.426·21-s − 1.37·22-s − 1.13·23-s − 1.88·24-s + 25-s + 0.709·26-s − 2.01·27-s − 0.119·28-s + 1.25·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2015\)    =    \(5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2015} (2014, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2015,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5421247023\)
\(L(\frac12)\)  \(\approx\)  \(0.5421247023\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;13,\;31\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{5,\;13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 + T \)
31 \( 1 + T \)
good2 \( 1 + 0.709T + T^{2} \)
3 \( 1 + 1.77T + T^{2} \)
7 \( 1 - 0.241T + T^{2} \)
11 \( 1 - 1.94T + T^{2} \)
17 \( 1 - 1.49T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.13T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.709T + T^{2} \)
47 \( 1 - 1.13T + T^{2} \)
53 \( 1 - 1.94T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.94T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 0.241T + T^{2} \)
97 \( 1 - 1.77T + T^{2} \)
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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

−9.528874280167046422022864038961, −8.884948165577991964994463960330, −7.58316067493780853719678788404, −6.97665717766408960704612331112, −6.00638195981166388042531833039, −5.54184251270767931524445381287, −4.67279901674174423321318360159, −3.88693579701816528991417045468, −1.77262720499890806050144658160, −0.976066662750734410475717357659, 0.976066662750734410475717357659, 1.77262720499890806050144658160, 3.88693579701816528991417045468, 4.67279901674174423321318360159, 5.54184251270767931524445381287, 6.00638195981166388042531833039, 6.97665717766408960704612331112, 7.58316067493780853719678788404, 8.884948165577991964994463960330, 9.528874280167046422022864038961

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