Properties

Label 2-2015-2015.1239-c0-0-3
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

Related objects

Downloads

Learn more

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: $\chi_{2015} (1239, \cdot )$
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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