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  − 1.49·2-s − 1.13·3-s + 1.24·4-s + 5-s + 1.70·6-s − 1.94·7-s − 0.360·8-s + 0.290·9-s − 1.49·10-s − 1.77·11-s − 1.41·12-s − 13-s + 2.90·14-s − 1.13·15-s − 0.700·16-s − 0.241·17-s − 0.435·18-s + 1.24·20-s + 2.20·21-s + 2.65·22-s + 0.709·23-s + 0.410·24-s + 25-s + 1.49·26-s + 0.805·27-s − 2.41·28-s + 1.70·30-s + ⋯
L(s)  = 1  − 1.49·2-s − 1.13·3-s + 1.24·4-s + 5-s + 1.70·6-s − 1.94·7-s − 0.360·8-s + 0.290·9-s − 1.49·10-s − 1.77·11-s − 1.41·12-s − 13-s + 2.90·14-s − 1.13·15-s − 0.700·16-s − 0.241·17-s − 0.435·18-s + 1.24·20-s + 2.20·21-s + 2.65·22-s + 0.709·23-s + 0.410·24-s + 25-s + 1.49·26-s + 0.805·27-s − 2.41·28-s + 1.70·30-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  =  \((2,\ 2015,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.1749855222\)
\(L(\frac12)\)  \(\approx\)  \(0.1749855222\)
\(L(1)\)   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 \( 1 + 1.49T + T^{2} \)
3 \( 1 + 1.13T + T^{2} \)
7 \( 1 + 1.94T + T^{2} \)
11 \( 1 + 1.77T + T^{2} \)
17 \( 1 + 0.241T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 0.709T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.49T + T^{2} \)
47 \( 1 + 0.709T + T^{2} \)
53 \( 1 + 1.77T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.77T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.94T + T^{2} \)
97 \( 1 - 1.13T + T^{2} \)
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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

−9.566026723900644548856258782567, −8.881344831898828691052287078353, −7.72936465237994315378142119415, −6.96853065697892845475933890276, −6.37650935054608422007155379696, −5.58247255859264136868634585543, −4.88061879317431361335496258371, −3.00319639382353361680295155149, −2.28209804667836791279486520083, −0.50326640238114147658436930821, 0.50326640238114147658436930821, 2.28209804667836791279486520083, 3.00319639382353361680295155149, 4.88061879317431361335496258371, 5.58247255859264136868634585543, 6.37650935054608422007155379696, 6.96853065697892845475933890276, 7.72936465237994315378142119415, 8.881344831898828691052287078353, 9.566026723900644548856258782567

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