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Properties

Label 2-15-15.2-c11-0-13
Degree $2$
Conductor $15$
Sign $0.969 + 0.244i$
Analytic cond. $11.5251$
Root an. cond. $3.39487$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (26.1 − 26.1i)2-s + (389. + 159. i)3-s + 681. i·4-s + (2.80e3 − 6.40e3i)5-s + (1.43e4 − 6.02e3i)6-s + (5.51e3 + 5.51e3i)7-s + (7.13e4 + 7.13e4i)8-s + (1.26e5 + 1.24e5i)9-s + (−9.41e4 − 2.40e5i)10-s + 2.13e5i·11-s + (−1.08e5 + 2.65e5i)12-s + (1.40e6 − 1.40e6i)13-s + 2.88e5·14-s + (2.11e6 − 2.04e6i)15-s + 2.33e6·16-s + (−2.24e6 + 2.24e6i)17-s + ⋯
L(s)  = 1  + (0.577 − 0.577i)2-s + (0.925 + 0.378i)3-s + 0.332i·4-s + (0.400 − 0.916i)5-s + (0.753 − 0.316i)6-s + (0.123 + 0.123i)7-s + (0.769 + 0.769i)8-s + (0.713 + 0.700i)9-s + (−0.297 − 0.760i)10-s + 0.398i·11-s + (−0.125 + 0.308i)12-s + (1.05 − 1.05i)13-s + 0.143·14-s + (0.717 − 0.696i)15-s + 0.556·16-s + (−0.383 + 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.969 + 0.244i$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ 0.969 + 0.244i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.52569 - 0.437372i\)
\(L(\frac12)\) \(\approx\) \(3.52569 - 0.437372i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-389. - 159. i)T \)
5 \( 1 + (-2.80e3 + 6.40e3i)T \)
good2 \( 1 + (-26.1 + 26.1i)T - 2.04e3iT^{2} \)
7 \( 1 + (-5.51e3 - 5.51e3i)T + 1.97e9iT^{2} \)
11 \( 1 - 2.13e5iT - 2.85e11T^{2} \)
13 \( 1 + (-1.40e6 + 1.40e6i)T - 1.79e12iT^{2} \)
17 \( 1 + (2.24e6 - 2.24e6i)T - 3.42e13iT^{2} \)
19 \( 1 + 1.02e7iT - 1.16e14T^{2} \)
23 \( 1 + (-1.46e7 - 1.46e7i)T + 9.52e14iT^{2} \)
29 \( 1 + 1.52e8T + 1.22e16T^{2} \)
31 \( 1 + 2.64e8T + 2.54e16T^{2} \)
37 \( 1 + (-2.07e8 - 2.07e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 1.29e8iT - 5.50e17T^{2} \)
43 \( 1 + (2.00e8 - 2.00e8i)T - 9.29e17iT^{2} \)
47 \( 1 + (9.40e8 - 9.40e8i)T - 2.47e18iT^{2} \)
53 \( 1 + (3.25e9 + 3.25e9i)T + 9.26e18iT^{2} \)
59 \( 1 - 4.71e9T + 3.01e19T^{2} \)
61 \( 1 + 4.47e9T + 4.35e19T^{2} \)
67 \( 1 + (1.40e10 + 1.40e10i)T + 1.22e20iT^{2} \)
71 \( 1 - 9.83e9iT - 2.31e20T^{2} \)
73 \( 1 + (-2.37e10 + 2.37e10i)T - 3.13e20iT^{2} \)
79 \( 1 - 6.03e9iT - 7.47e20T^{2} \)
83 \( 1 + (-3.18e10 - 3.18e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 9.38e10T + 2.77e21T^{2} \)
97 \( 1 + (1.56e10 + 1.56e10i)T + 7.15e21iT^{2} \)
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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

−16.44277033623562698067693358185, −15.05246868872852087426005578367, −13.36343486015740697562691592943, −12.91111285600410837364270991085, −10.99233475505639688132725743389, −9.184996022128620226837737450756, −7.962369991635659133702611244267, −5.01370398210804073040554050072, −3.52783101563191880139400354881, −1.84115423520032106741105842906, 1.71646251582992005530333895976, 3.77143896940361358887400800665, 6.11250450825879885129225944946, 7.29439180669736350717220054593, 9.278049400838950593392872817167, 10.91177897331110731100933200169, 13.23100653483383325197676038530, 14.14252910652270321431367631076, 14.87605514876592755782223561688, 16.28624346344620444092332324485

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