Properties

Label 2-15-15.2-c9-0-7
Degree $2$
Conductor $15$
Sign $0.710 - 0.703i$
Analytic cond. $7.72553$
Root an. cond. $2.77948$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.02 − 4.02i)2-s + (130. − 52.2i)3-s + 479. i·4-s + (−216. + 1.38e3i)5-s + (313. − 734. i)6-s + (17.2 + 17.2i)7-s + (3.99e3 + 3.99e3i)8-s + (1.42e4 − 1.36e4i)9-s + (4.69e3 + 6.42e3i)10-s + 8.84e4i·11-s + (2.50e4 + 6.24e4i)12-s + (1.06e5 − 1.06e5i)13-s + 139.·14-s + (4.40e4 + 1.91e5i)15-s − 2.13e5·16-s + (2.21e5 − 2.21e5i)17-s + ⋯
L(s)  = 1  + (0.177 − 0.177i)2-s + (0.927 − 0.372i)3-s + 0.936i·4-s + (−0.154 + 0.987i)5-s + (0.0988 − 0.231i)6-s + (0.00272 + 0.00272i)7-s + (0.344 + 0.344i)8-s + (0.722 − 0.691i)9-s + (0.148 + 0.203i)10-s + 1.82i·11-s + (0.349 + 0.869i)12-s + (1.03 − 1.03i)13-s + 0.000968·14-s + (0.224 + 0.974i)15-s − 0.813·16-s + (0.641 − 0.641i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.710 - 0.703i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ 0.710 - 0.703i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.21475 + 0.910821i\)
\(L(\frac12)\) \(\approx\) \(2.21475 + 0.910821i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-130. + 52.2i)T \)
5 \( 1 + (216. - 1.38e3i)T \)
good2 \( 1 + (-4.02 + 4.02i)T - 512iT^{2} \)
7 \( 1 + (-17.2 - 17.2i)T + 4.03e7iT^{2} \)
11 \( 1 - 8.84e4iT - 2.35e9T^{2} \)
13 \( 1 + (-1.06e5 + 1.06e5i)T - 1.06e10iT^{2} \)
17 \( 1 + (-2.21e5 + 2.21e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 1.38e5iT - 3.22e11T^{2} \)
23 \( 1 + (1.10e6 + 1.10e6i)T + 1.80e12iT^{2} \)
29 \( 1 + 2.89e6T + 1.45e13T^{2} \)
31 \( 1 - 3.99e6T + 2.64e13T^{2} \)
37 \( 1 + (-4.00e6 - 4.00e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 9.96e6iT - 3.27e14T^{2} \)
43 \( 1 + (-1.56e6 + 1.56e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (-1.81e7 + 1.81e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (1.37e7 + 1.37e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 7.44e7T + 8.66e15T^{2} \)
61 \( 1 - 1.28e8T + 1.16e16T^{2} \)
67 \( 1 + (-7.61e7 - 7.61e7i)T + 2.72e16iT^{2} \)
71 \( 1 - 3.68e7iT - 4.58e16T^{2} \)
73 \( 1 + (1.01e8 - 1.01e8i)T - 5.88e16iT^{2} \)
79 \( 1 + 3.01e8iT - 1.19e17T^{2} \)
83 \( 1 + (-1.97e7 - 1.97e7i)T + 1.86e17iT^{2} \)
89 \( 1 + 7.06e8T + 3.50e17T^{2} \)
97 \( 1 + (1.62e8 + 1.62e8i)T + 7.60e17iT^{2} \)
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

−17.72791702883448145426428414959, −15.69071774352223932558891140085, −14.53062678048916572529381642375, −13.17552184857874914909197896667, −12.01574092905133933592118838610, −10.07359472139928020937154161360, −8.113049370389857672216705463602, −7.05665021416247457618843862189, −3.80286312376229969073215396079, −2.39668742211310193480209892455, 1.28302946908350242058187234153, 3.97741747306831090473990252140, 5.81494680039744145015480853501, 8.317191614259296237759304665132, 9.441587051311520291347933075944, 11.14576346689970984912663461093, 13.39494003028477079249239530640, 14.14770248500988537299035312291, 15.72627081461203794425451633279, 16.44792762735158753235514206869

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