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  − 1.94·2-s − 1.49·3-s + 2.77·4-s + 5-s + 2.90·6-s + 1.13·7-s − 3.43·8-s + 1.24·9-s − 1.94·10-s − 0.709·11-s − 4.14·12-s + 13-s − 2.20·14-s − 1.49·15-s + 3.90·16-s + 1.77·17-s − 2.41·18-s + 2.77·20-s − 1.70·21-s + 1.37·22-s + 0.241·23-s + 5.14·24-s + 25-s − 1.94·26-s − 0.360·27-s + 3.14·28-s + 2.90·30-s + ⋯
L(s)  = 1  − 1.94·2-s − 1.49·3-s + 2.77·4-s + 5-s + 2.90·6-s + 1.13·7-s − 3.43·8-s + 1.24·9-s − 1.94·10-s − 0.709·11-s − 4.14·12-s + 13-s − 2.20·14-s − 1.49·15-s + 3.90·16-s + 1.77·17-s − 2.41·18-s + 2.77·20-s − 1.70·21-s + 1.37·22-s + 0.241·23-s + 5.14·24-s + 25-s − 1.94·26-s − 0.360·27-s + 3.14·28-s + 2.90·30-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  =  \((2,\ 2015,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.4481774745\)
\(L(\frac12)\)  \(\approx\)  \(0.4481774745\)
\(L(1)\)   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 \( 1 + 1.94T + T^{2} \)
3 \( 1 + 1.49T + T^{2} \)
7 \( 1 - 1.13T + T^{2} \)
11 \( 1 + 0.709T + T^{2} \)
17 \( 1 - 1.77T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 0.241T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.94T + T^{2} \)
47 \( 1 - 0.241T + T^{2} \)
53 \( 1 + 0.709T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.709T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.13T + T^{2} \)
97 \( 1 + 1.49T + T^{2} \)
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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

−9.558694572938584462409593386437, −8.450724256933496999661895032268, −8.019150457135728814287824353660, −7.06211980277848543889074864226, −6.27619253777148262097435373925, −5.66841373458882716898087155983, −5.01457607852384019771441982497, −3.01900977791589968834662152145, −1.66573551457843871328628517291, −1.05645648762942095662793994098, 1.05645648762942095662793994098, 1.66573551457843871328628517291, 3.01900977791589968834662152145, 5.01457607852384019771441982497, 5.66841373458882716898087155983, 6.27619253777148262097435373925, 7.06211980277848543889074864226, 8.019150457135728814287824353660, 8.450724256933496999661895032268, 9.558694572938584462409593386437

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