Properties

Label 2-2015-2015.2014-c0-0-15
Degree $2$
Conductor $2015$
Sign $1$
Analytic cond. $1.00561$
Root an. cond. $1.00280$
Motivic weight $0$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $1$
Analytic conductor: \(1.00561\)
Root analytic conductor: \(1.00280\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (2014, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2015,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.103658311\)
\(L(\frac12)\) \(\approx\) \(2.103658311\)
\(L(1)\) 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 \( 1 - 1.94T + T^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.748095285475432009856871183692, −8.177984672036051286615261573008, −7.11461840075603576610267928701, −6.66615138907150370775517314705, −5.96998792872770954554930417310, −5.38913875767205262704620256587, −4.53252905307959028205511788791, −3.55247825461183670086020724073, −3.30138835769471909622945047789, −1.29692522388745557124190793142, 1.29692522388745557124190793142, 3.30138835769471909622945047789, 3.55247825461183670086020724073, 4.53252905307959028205511788791, 5.38913875767205262704620256587, 5.96998792872770954554930417310, 6.66615138907150370775517314705, 7.11461840075603576610267928701, 8.177984672036051286615261573008, 9.748095285475432009856871183692

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