Properties

Label 2-2015-2015.2014-c0-0-5
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  − 1.13·2-s − 1.94·3-s + 0.290·4-s − 5-s + 2.20·6-s + 1.49·7-s + 0.805·8-s + 2.77·9-s + 1.13·10-s − 0.241·11-s − 0.564·12-s + 13-s − 1.70·14-s + 1.94·15-s − 1.20·16-s − 0.709·17-s − 3.14·18-s − 0.290·20-s − 2.90·21-s + 0.273·22-s + 1.77·23-s − 1.56·24-s + 25-s − 1.13·26-s − 3.43·27-s + 0.435·28-s − 2.20·30-s + ⋯
L(s)  = 1  − 1.13·2-s − 1.94·3-s + 0.290·4-s − 5-s + 2.20·6-s + 1.49·7-s + 0.805·8-s + 2.77·9-s + 1.13·10-s − 0.241·11-s − 0.564·12-s + 13-s − 1.70·14-s + 1.94·15-s − 1.20·16-s − 0.709·17-s − 3.14·18-s − 0.290·20-s − 2.90·21-s + 0.273·22-s + 1.77·23-s − 1.56·24-s + 25-s − 1.13·26-s − 3.43·27-s + 0.435·28-s − 2.20·30-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

\(L(\frac{1}{2})\) \(\approx\) \(0.3154855530\)
\(L(\frac12)\) \(\approx\) \(0.3154855530\)
\(L(1)\) 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 \( 1 + 1.13T + T^{2} \)
3 \( 1 + 1.94T + T^{2} \)
7 \( 1 - 1.49T + T^{2} \)
11 \( 1 + 0.241T + T^{2} \)
17 \( 1 + 0.709T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.77T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.13T + T^{2} \)
47 \( 1 + 1.77T + T^{2} \)
53 \( 1 - 0.241T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.241T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.49T + T^{2} \)
97 \( 1 - 1.94T + T^{2} \)
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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

−9.248726736755246511167251862689, −8.564338434828315120514522737096, −7.67990712680331224039440377427, −7.22291343623956713816771762328, −6.34354510130673875553621904533, −5.10210854754594609853561161621, −4.79343622931051103821214072591, −3.92253042532431160074776311965, −1.64585215054058414052770944440, −0.78841751567386737534281202557, 0.78841751567386737534281202557, 1.64585215054058414052770944440, 3.92253042532431160074776311965, 4.79343622931051103821214072591, 5.10210854754594609853561161621, 6.34354510130673875553621904533, 7.22291343623956713816771762328, 7.67990712680331224039440377427, 8.564338434828315120514522737096, 9.248726736755246511167251862689

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