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Properties

Label 2-15-15.8-c11-0-10
Degree $2$
Conductor $15$
Sign $-0.330 - 0.943i$
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  + (61.8 + 61.8i)2-s + (420. + 3.69i)3-s + 5.60e3i·4-s + (4.34e3 − 5.47e3i)5-s + (2.58e4 + 2.62e4i)6-s + (−1.17e4 + 1.17e4i)7-s + (−2.20e5 + 2.20e5i)8-s + (1.77e5 + 3.11e3i)9-s + (6.07e5 − 6.94e4i)10-s − 4.35e5i·11-s + (−2.07e4 + 2.36e6i)12-s + (−4.68e5 − 4.68e5i)13-s − 1.44e6·14-s + (1.85e6 − 2.28e6i)15-s − 1.57e7·16-s + (−2.28e6 − 2.28e6i)17-s + ⋯
L(s)  = 1  + (1.36 + 1.36i)2-s + (0.999 + 0.00878i)3-s + 2.73i·4-s + (0.622 − 0.782i)5-s + (1.35 + 1.37i)6-s + (−0.263 + 0.263i)7-s + (−2.37 + 2.37i)8-s + (0.999 + 0.0175i)9-s + (1.92 − 0.219i)10-s − 0.815i·11-s + (−0.0240 + 2.73i)12-s + (−0.350 − 0.350i)13-s − 0.720·14-s + (0.629 − 0.777i)15-s − 3.76·16-s + (−0.390 − 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.330 - 0.943i$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ -0.330 - 0.943i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.83006 + 3.98995i\)
\(L(\frac12)\) \(\approx\) \(2.83006 + 3.98995i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-420. - 3.69i)T \)
5 \( 1 + (-4.34e3 + 5.47e3i)T \)
good2 \( 1 + (-61.8 - 61.8i)T + 2.04e3iT^{2} \)
7 \( 1 + (1.17e4 - 1.17e4i)T - 1.97e9iT^{2} \)
11 \( 1 + 4.35e5iT - 2.85e11T^{2} \)
13 \( 1 + (4.68e5 + 4.68e5i)T + 1.79e12iT^{2} \)
17 \( 1 + (2.28e6 + 2.28e6i)T + 3.42e13iT^{2} \)
19 \( 1 + 1.22e6iT - 1.16e14T^{2} \)
23 \( 1 + (1.59e7 - 1.59e7i)T - 9.52e14iT^{2} \)
29 \( 1 - 1.86e8T + 1.22e16T^{2} \)
31 \( 1 + 3.77e7T + 2.54e16T^{2} \)
37 \( 1 + (-3.83e8 + 3.83e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 5.80e8iT - 5.50e17T^{2} \)
43 \( 1 + (-1.11e8 - 1.11e8i)T + 9.29e17iT^{2} \)
47 \( 1 + (3.45e8 + 3.45e8i)T + 2.47e18iT^{2} \)
53 \( 1 + (1.15e9 - 1.15e9i)T - 9.26e18iT^{2} \)
59 \( 1 + 8.18e9T + 3.01e19T^{2} \)
61 \( 1 + 1.00e10T + 4.35e19T^{2} \)
67 \( 1 + (1.32e9 - 1.32e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 3.74e9iT - 2.31e20T^{2} \)
73 \( 1 + (5.37e9 + 5.37e9i)T + 3.13e20iT^{2} \)
79 \( 1 + 1.68e10iT - 7.47e20T^{2} \)
83 \( 1 + (-6.15e9 + 6.15e9i)T - 1.28e21iT^{2} \)
89 \( 1 - 1.64e10T + 2.77e21T^{2} \)
97 \( 1 + (5.60e10 - 5.60e10i)T - 7.15e21iT^{2} \)
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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

−16.41049324817983901122326396048, −15.56410598660796234714489968249, −14.19139612287520937815235752460, −13.42401069431112222370696806348, −12.38882009460787494026677439005, −9.082445160155168299732061963850, −7.87613534087813755430339521575, −6.13384774531407471367789796502, −4.60221335549185434181807001132, −2.88419020085432949671178231954, 1.80631622699547426570553905041, 2.91633315259587506689594936237, 4.40988262325249405097067631160, 6.57926993297366341867499189432, 9.647789049297808994378829241075, 10.52216626592601934969372583322, 12.32967679936517758719835391768, 13.52440813902223755767761212777, 14.35834601879755350228019150259, 15.30173311095508073840806882601

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