Properties

Degree $2$
Conductor $2015$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.77·2-s − 0.241·3-s + 2.13·4-s + 5-s − 0.426·6-s − 0.709·7-s + 2.01·8-s − 0.941·9-s + 1.77·10-s + 1.49·11-s − 0.514·12-s − 13-s − 1.25·14-s − 0.241·15-s + 1.42·16-s − 1.13·17-s − 1.66·18-s + 2.13·20-s + 0.170·21-s + 2.65·22-s + 1.94·23-s − 0.485·24-s + 25-s − 1.77·26-s + 0.468·27-s − 1.51·28-s − 0.426·30-s + ⋯
L(s)  = 1  + 1.77·2-s − 0.241·3-s + 2.13·4-s + 5-s − 0.426·6-s − 0.709·7-s + 2.01·8-s − 0.941·9-s + 1.77·10-s + 1.49·11-s − 0.514·12-s − 13-s − 1.25·14-s − 0.241·15-s + 1.42·16-s − 1.13·17-s − 1.66·18-s + 2.13·20-s + 0.170·21-s + 2.65·22-s + 1.94·23-s − 0.485·24-s + 25-s − 1.77·26-s + 0.468·27-s − 1.51·28-s − 0.426·30-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

Degree: \(2\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{2015} (2014, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2015,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.139761479\)
\(L(\frac12)\) \(\approx\) \(3.139761479\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 + T \)
31 \( 1 + T \)
good2 \( 1 - 1.77T + T^{2} \)
3 \( 1 + 0.241T + T^{2} \)
7 \( 1 + 0.709T + T^{2} \)
11 \( 1 - 1.49T + T^{2} \)
17 \( 1 + 1.13T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.94T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.77T + T^{2} \)
47 \( 1 + 1.94T + T^{2} \)
53 \( 1 - 1.49T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.49T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.709T + T^{2} \)
97 \( 1 - 0.241T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.292014391658777856033638687805, −8.807392771275278954847223213749, −7.09007587341198809330522714171, −6.63488100521241398194754028949, −6.12939132372275766378745388106, −5.19435538235853106664368840107, −4.71595475025130215550375973528, −3.48256531394049166765026733114, −2.83489815514316088580991592171, −1.81549620389973541533802470866, 1.81549620389973541533802470866, 2.83489815514316088580991592171, 3.48256531394049166765026733114, 4.71595475025130215550375973528, 5.19435538235853106664368840107, 6.12939132372275766378745388106, 6.63488100521241398194754028949, 7.09007587341198809330522714171, 8.807392771275278954847223213749, 9.292014391658777856033638687805

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