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Properties

Label 2-15-15.8-c9-0-4
Degree $2$
Conductor $15$
Sign $0.0164 - 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.02 − 4.02i)2-s + (52.2 + 130. i)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 + (6.24e4 − 2.50e4i)12-s + (1.06e5 + 1.06e5i)13-s − 139.·14-s + (−1.68e5 + 1.00e5i)15-s − 2.13e5·16-s + (−2.21e5 − 2.21e5i)17-s + ⋯
L(s)  = 1  + (−0.177 − 0.177i)2-s + (0.372 + 0.927i)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 + (0.869 − 0.349i)12-s + (1.03 + 1.03i)13-s − 0.000968·14-s + (−0.859 + 0.511i)15-s − 0.813·16-s + (−0.641 − 0.641i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0164 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0164 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.0164 - 0.999i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ 0.0164 - 0.999i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.08591 + 1.06821i\)
\(L(\frac12)\) \(\approx\) \(1.08591 + 1.06821i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-52.2 - 130. i)T \)
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 \( 1 + (1.62e8 - 1.62e8i)T - 7.60e17iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.71459445822544581693879160210, −15.75598417485940116231561051425, −14.86931102871041278700384860914, −13.88716075434112016285684637018, −11.31330556419898110730860517759, −10.23504283402201712128335345892, −9.175854316345399978222329190241, −6.68166445748887887313961675183, −4.59004873859215511475293616061, −2.29067995393657131114234456319, 0.844015378544759728355817349680, 3.29783526035994602497984975930, 6.09637974724769872350191101999, 8.153908364490491208775668642646, 8.707546292390439511301638835245, 11.48752556366599375797829136069, 12.97098313328668533340236622928, 13.53576874737907336118340694928, 15.78176139211178271780616996581, 16.99564318885206796428246240442

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