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Properties

Label 2-15-15.2-c11-0-0
Degree $2$
Conductor $15$
Sign $-0.595 - 0.803i$
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  + (1.50 − 1.50i)2-s + (−221. − 357. i)3-s + 2.04e3i·4-s + (3.16e3 − 6.22e3i)5-s + (−873. − 205. i)6-s + (−3.11e4 − 3.11e4i)7-s + (6.16e3 + 6.16e3i)8-s + (−7.89e4 + 1.58e5i)9-s + (−4.61e3 − 1.41e4i)10-s + 7.09e5i·11-s + (7.31e5 − 4.52e5i)12-s + (−1.08e6 + 1.08e6i)13-s − 9.37e4·14-s + (−2.93e6 + 2.46e5i)15-s − 4.16e6·16-s + (−3.61e5 + 3.61e5i)17-s + ⋯
L(s)  = 1  + (0.0333 − 0.0333i)2-s + (−0.526 − 0.850i)3-s + 0.997i·4-s + (0.453 − 0.891i)5-s + (−0.0458 − 0.0107i)6-s + (−0.699 − 0.699i)7-s + (0.0665 + 0.0665i)8-s + (−0.445 + 0.895i)9-s + (−0.0145 − 0.0447i)10-s + 1.32i·11-s + (0.848 − 0.525i)12-s + (−0.807 + 0.807i)13-s − 0.0466·14-s + (−0.996 + 0.0837i)15-s − 0.993·16-s + (−0.0617 + 0.0617i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.595 - 0.803i$
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: \((2,\ 15,\ (\ :11/2),\ -0.595 - 0.803i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.159008 + 0.315608i\)
\(L(\frac12)\) \(\approx\) \(0.159008 + 0.315608i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (221. + 357. i)T \)
5 \( 1 + (-3.16e3 + 6.22e3i)T \)
good2 \( 1 + (-1.50 + 1.50i)T - 2.04e3iT^{2} \)
7 \( 1 + (3.11e4 + 3.11e4i)T + 1.97e9iT^{2} \)
11 \( 1 - 7.09e5iT - 2.85e11T^{2} \)
13 \( 1 + (1.08e6 - 1.08e6i)T - 1.79e12iT^{2} \)
17 \( 1 + (3.61e5 - 3.61e5i)T - 3.42e13iT^{2} \)
19 \( 1 - 1.54e7iT - 1.16e14T^{2} \)
23 \( 1 + (2.97e7 + 2.97e7i)T + 9.52e14iT^{2} \)
29 \( 1 + 2.75e7T + 1.22e16T^{2} \)
31 \( 1 - 1.35e8T + 2.54e16T^{2} \)
37 \( 1 + (1.65e7 + 1.65e7i)T + 1.77e17iT^{2} \)
41 \( 1 + 3.86e8iT - 5.50e17T^{2} \)
43 \( 1 + (1.00e9 - 1.00e9i)T - 9.29e17iT^{2} \)
47 \( 1 + (2.83e8 - 2.83e8i)T - 2.47e18iT^{2} \)
53 \( 1 + (8.90e8 + 8.90e8i)T + 9.26e18iT^{2} \)
59 \( 1 + 1.00e10T + 3.01e19T^{2} \)
61 \( 1 - 4.27e9T + 4.35e19T^{2} \)
67 \( 1 + (-1.45e9 - 1.45e9i)T + 1.22e20iT^{2} \)
71 \( 1 + 1.94e10iT - 2.31e20T^{2} \)
73 \( 1 + (-8.80e8 + 8.80e8i)T - 3.13e20iT^{2} \)
79 \( 1 - 2.87e10iT - 7.47e20T^{2} \)
83 \( 1 + (-4.46e10 - 4.46e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 5.35e10T + 2.77e21T^{2} \)
97 \( 1 + (6.90e10 + 6.90e10i)T + 7.15e21iT^{2} \)
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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

−16.97004530769593197051563652879, −16.46401798756705135740233691604, −13.89978887429378935760241523995, −12.64900175521648556803820003957, −12.13528751326032615402064514786, −9.869417063967081763878532317476, −7.966254660611539865122953118394, −6.64022252287957861911452055833, −4.47647911526384520777838202420, −1.94970375680645155218744555148, 0.15595658950118274760181657006, 2.94540826125170342263168988494, 5.40057017869985248328401938942, 6.35764930596341993199793678630, 9.328245842510576028849666761968, 10.33720520689113432232902044832, 11.48482683908194621971439067402, 13.69245171520537747788939613276, 15.07953938795151304124852088707, 15.82027844062566838158058951150

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