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Properties

Label 2-2015-2015.309-c0-0-0
Degree $2$
Conductor $2015$
Sign $0.271 - 0.962i$
Analytic cond. $1.00561$
Root an. cond. $1.00280$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)3-s + (0.5 − 0.866i)4-s + i·5-s + (−0.499 + 0.866i)9-s + 1.41·12-s + (−0.258 − 0.965i)13-s + (−1.22 + 0.707i)15-s + (−0.499 − 0.866i)16-s + (−0.965 + 1.67i)17-s + (0.866 + 0.5i)19-s + (0.866 + 0.5i)20-s + (0.965 + 1.67i)23-s − 25-s − i·31-s + (0.5 + 0.866i)36-s + (−0.448 + 0.258i)37-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)3-s + (0.5 − 0.866i)4-s + i·5-s + (−0.499 + 0.866i)9-s + 1.41·12-s + (−0.258 − 0.965i)13-s + (−1.22 + 0.707i)15-s + (−0.499 − 0.866i)16-s + (−0.965 + 1.67i)17-s + (0.866 + 0.5i)19-s + (0.866 + 0.5i)20-s + (0.965 + 1.67i)23-s − 25-s − i·31-s + (0.5 + 0.866i)36-s + (−0.448 + 0.258i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.271 - 0.962i$
Analytic conductor: \(1.00561\)
Root analytic conductor: \(1.00280\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2015,\ (\ :0),\ 0.271 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.611290062\)
\(L(\frac12)\) \(\approx\) \(1.611290062\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
13 \( 1 + (0.258 + 0.965i)T \)
31 \( 1 + iT \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 0.517T + T^{2} \)
59 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.93iT - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.571797382088429176130088445520, −9.063980510104068636390863966269, −7.86330369092436556790776897166, −7.28569215041651912289902916673, −6.11932303184167126686073946365, −5.63332737080760345800047366588, −4.53403026673984836331213938128, −3.54835996002656772173805227359, −2.92170488909230804235853265364, −1.80467498983473705864424113214, 1.15289720740958487103304703075, 2.39521511116765004852760180033, 2.85722622951202341065880455604, 4.30654970384145470941794363830, 4.98311959864351356443514054154, 6.45075963905246247102917530870, 7.06173686502054042608022955828, 7.49001578274797737999214838217, 8.448009795033291191074035493334, 8.907326762777257643749344359998

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