↑

Properties

Label 2-15-3.2-c10-0-12
Degree $2$
Conductor $15$
Sign $-0.587 - 0.809i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 52.0i·2-s + (196. − 142. i)3-s − 1.68e3·4-s + 1.39e3i·5-s + (−7.43e3 − 1.02e4i)6-s − 3.23e4·7-s + 3.46e4i·8-s + (1.82e4 − 5.61e4i)9-s + 7.28e4·10-s + 1.00e4i·11-s + (−3.32e5 + 2.41e5i)12-s − 1.86e3·13-s + 1.68e6i·14-s + (1.99e5 + 2.74e5i)15-s + 7.63e4·16-s − 1.52e6i·17-s + ⋯
L(s)  = 1  − 1.62i·2-s + (0.809 − 0.587i)3-s − 1.65·4-s + 0.447i·5-s + (−0.956 − 1.31i)6-s − 1.92·7-s + 1.05i·8-s + (0.309 − 0.950i)9-s + 0.728·10-s + 0.0622i·11-s + (−1.33 + 0.969i)12-s − 0.00502·13-s + 3.13i·14-s + (0.262 + 0.361i)15-s + 0.0728·16-s − 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.587 - 0.809i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ -0.587 - 0.809i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.534300 + 1.04841i\)
\(L(\frac12)\) \(\approx\) \(0.534300 + 1.04841i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-196. + 142. i)T \)
5 \( 1 - 1.39e3iT \)
good2 \( 1 + 52.0iT - 1.02e3T^{2} \)
7 \( 1 + 3.23e4T + 2.82e8T^{2} \)
11 \( 1 - 1.00e4iT - 2.59e10T^{2} \)
13 \( 1 + 1.86e3T + 1.37e11T^{2} \)
17 \( 1 + 1.52e6iT - 2.01e12T^{2} \)
19 \( 1 - 3.08e6T + 6.13e12T^{2} \)
23 \( 1 + 8.67e6iT - 4.14e13T^{2} \)
29 \( 1 + 7.01e6iT - 4.20e14T^{2} \)
31 \( 1 + 7.59e6T + 8.19e14T^{2} \)
37 \( 1 + 9.52e7T + 4.80e15T^{2} \)
41 \( 1 - 6.27e7iT - 1.34e16T^{2} \)
43 \( 1 - 7.08e7T + 2.16e16T^{2} \)
47 \( 1 + 3.66e7iT - 5.25e16T^{2} \)
53 \( 1 + 2.99e8iT - 1.74e17T^{2} \)
59 \( 1 - 2.34e8iT - 5.11e17T^{2} \)
61 \( 1 - 7.97e8T + 7.13e17T^{2} \)
67 \( 1 - 7.89e8T + 1.82e18T^{2} \)
71 \( 1 + 2.27e9iT - 3.25e18T^{2} \)
73 \( 1 - 6.29e8T + 4.29e18T^{2} \)
79 \( 1 + 3.79e9T + 9.46e18T^{2} \)
83 \( 1 + 9.37e8iT - 1.55e19T^{2} \)
89 \( 1 - 6.91e9iT - 3.11e19T^{2} \)
97 \( 1 - 7.43e9T + 7.37e19T^{2} \)
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

−15.93453393787856024489082430771, −13.96205700602533922687246339928, −12.99306566899432104492389229039, −11.98681803370202220382115040464, −10.09702164195548045634636398476, −9.212363237028885999035916318870, −6.85685201778637988376481148997, −3.47861039855908001866821073917, −2.59404271111894946615430764259, −0.50575033695799381683950401532, 3.60794584775113421466571145221, 5.61954818231636696756231362649, 7.25532701251867681771589614964, 8.830192441059221488021448317776, 9.822976051160328511706044784786, 12.96986714420363640062854617321, 13.99706220744971767781140797564, 15.57536654819822858656165395778, 16.00280241649059835219368553414, 17.12712688286238351374707425864

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