↑

Properties

Label 2-30-15.14-c10-0-18
Degree $2$
Conductor $30$
Sign $-0.466 + 0.884i$
Analytic cond. $19.0607$
Root an. cond. $4.36585$
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  + 22.6·2-s + (161. − 181. i)3-s + 512.·4-s + (−3.03e3 + 749. i)5-s + (3.65e3 − 4.10e3i)6-s − 5.24e3i·7-s + 1.15e4·8-s + (−6.81e3 − 5.86e4i)9-s + (−6.86e4 + 1.69e4i)10-s − 2.60e5i·11-s + (8.27e4 − 9.29e4i)12-s − 1.16e5i·13-s − 1.18e5i·14-s + (−3.54e5 + 6.71e5i)15-s + 2.62e5·16-s − 1.36e6·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.665 − 0.746i)3-s + 0.500·4-s + (−0.970 + 0.239i)5-s + (0.470 − 0.528i)6-s − 0.311i·7-s + 0.353·8-s + (−0.115 − 0.993i)9-s + (−0.686 + 0.169i)10-s − 1.61i·11-s + (0.332 − 0.373i)12-s − 0.313i·13-s − 0.220i·14-s + (−0.466 + 0.884i)15-s + 0.250·16-s − 0.962·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $-0.466 + 0.884i$
Analytic conductor: \(19.0607\)
Root analytic conductor: \(4.36585\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :5),\ -0.466 + 0.884i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.31097 - 2.17340i\)
\(L(\frac12)\) \(\approx\) \(1.31097 - 2.17340i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 22.6T \)
3 \( 1 + (-161. + 181. i)T \)
5 \( 1 + (3.03e3 - 749. i)T \)
good7 \( 1 + 5.24e3iT - 2.82e8T^{2} \)
11 \( 1 + 2.60e5iT - 2.59e10T^{2} \)
13 \( 1 + 1.16e5iT - 1.37e11T^{2} \)
17 \( 1 + 1.36e6T + 2.01e12T^{2} \)
19 \( 1 - 2.35e5T + 6.13e12T^{2} \)
23 \( 1 + 3.46e6T + 4.14e13T^{2} \)
29 \( 1 - 9.31e5iT - 4.20e14T^{2} \)
31 \( 1 - 2.67e7T + 8.19e14T^{2} \)
37 \( 1 + 1.03e8iT - 4.80e15T^{2} \)
41 \( 1 - 6.75e7iT - 1.34e16T^{2} \)
43 \( 1 - 1.84e8iT - 2.16e16T^{2} \)
47 \( 1 - 2.26e8T + 5.25e16T^{2} \)
53 \( 1 - 1.76e7T + 1.74e17T^{2} \)
59 \( 1 - 1.35e9iT - 5.11e17T^{2} \)
61 \( 1 - 9.20e8T + 7.13e17T^{2} \)
67 \( 1 - 9.32e8iT - 1.82e18T^{2} \)
71 \( 1 + 2.65e9iT - 3.25e18T^{2} \)
73 \( 1 + 3.70e9iT - 4.29e18T^{2} \)
79 \( 1 - 4.20e9T + 9.46e18T^{2} \)
83 \( 1 + 7.29e9T + 1.55e19T^{2} \)
89 \( 1 - 3.48e9iT - 3.11e19T^{2} \)
97 \( 1 - 9.21e9iT - 7.37e19T^{2} \)
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

−14.12636257134008638038642828306, −13.25971355953442373656181353269, −11.96803091924622384842342545033, −10.89966608310170696308515947504, −8.636817116442101182482346280607, −7.53179206075914626130263126715, −6.18885773411723942336424492347, −3.97258492493574809385112378997, −2.79390081184084053943011482312, −0.67466842018283927514430582639, 2.27953160779938569106654945259, 3.96561030809128263912537506872, 4.86255115482486625841268933143, 7.13312456859449145679594064939, 8.536253863781202188703249766295, 10.01987527848247843055286190232, 11.52386769747701841558442455660, 12.65058008627879547686583227814, 14.06865426366717044811949608540, 15.40331160939746668353499417854

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