Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.241·2-s + 0.709·3-s − 0.941·4-s − 5-s − 0.170·6-s − 1.77·7-s + 0.468·8-s − 0.497·9-s + 0.241·10-s + 1.13·11-s − 0.667·12-s − 13-s + 0.426·14-s − 0.709·15-s + 0.829·16-s + 1.94·17-s + 0.119·18-s + 0.941·20-s − 1.25·21-s − 0.273·22-s + 1.49·23-s + 0.332·24-s + 25-s + 0.241·26-s − 1.06·27-s + 1.66·28-s + 0.170·30-s + ⋯
L(s)  = 1  − 0.241·2-s + 0.709·3-s − 0.941·4-s − 5-s − 0.170·6-s − 1.77·7-s + 0.468·8-s − 0.497·9-s + 0.241·10-s + 1.13·11-s − 0.667·12-s − 13-s + 0.426·14-s − 0.709·15-s + 0.829·16-s + 1.94·17-s + 0.119·18-s + 0.941·20-s − 1.25·21-s − 0.273·22-s + 1.49·23-s + 0.332·24-s + 25-s + 0.241·26-s − 1.06·27-s + 1.66·28-s + 0.170·30-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

\(L(\frac{1}{2})\) \(\approx\) \(0.6754621010\)
\(L(\frac12)\) \(\approx\) \(0.6754621010\)
\(L(1)\) 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 \( 1 + 0.241T + T^{2} \)
3 \( 1 - 0.709T + T^{2} \)
7 \( 1 + 1.77T + T^{2} \)
11 \( 1 - 1.13T + T^{2} \)
17 \( 1 - 1.94T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.49T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 0.241T + T^{2} \)
47 \( 1 - 1.49T + T^{2} \)
53 \( 1 + 1.13T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.13T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.77T + T^{2} \)
97 \( 1 - 0.709T + T^{2} \)
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

−9.231128206993928047183578181308, −8.787559286829685999830269067092, −7.81347785847448157897884612512, −7.26801640739764489979896167791, −6.28331138789285457717973570435, −5.25504363970816139280431196278, −4.19273826275226385932867657323, −3.32605393727601151168199851474, −3.02315865078478646114561752633, −0.812816826684260909474368882356, 0.812816826684260909474368882356, 3.02315865078478646114561752633, 3.32605393727601151168199851474, 4.19273826275226385932867657323, 5.25504363970816139280431196278, 6.28331138789285457717973570435, 7.26801640739764489979896167791, 7.81347785847448157897884612512, 8.787559286829685999830269067092, 9.231128206993928047183578181308

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