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Properties

Label 2-15-15.8-c9-0-14
Degree $2$
Conductor $15$
Sign $-0.542 - 0.840i$
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  + (−13.8 − 13.8i)2-s + (−55.7 − 128. i)3-s − 127. i·4-s + (−919. − 1.05e3i)5-s + (−1.01e3 + 2.55e3i)6-s + (6.79e3 − 6.79e3i)7-s + (−8.86e3 + 8.86e3i)8-s + (−1.34e4 + 1.43e4i)9-s + (−1.83e3 + 2.73e4i)10-s + 2.84e4i·11-s + (−1.63e4 + 7.07e3i)12-s + (2.82e4 + 2.82e4i)13-s − 1.88e5·14-s + (−8.41e4 + 1.77e5i)15-s + 1.80e5·16-s + (2.11e4 + 2.11e4i)17-s + ⋯
L(s)  = 1  + (−0.613 − 0.613i)2-s + (−0.397 − 0.917i)3-s − 0.248i·4-s + (−0.658 − 0.752i)5-s + (−0.319 + 0.806i)6-s + (1.07 − 1.07i)7-s + (−0.765 + 0.765i)8-s + (−0.684 + 0.728i)9-s + (−0.0579 + 0.865i)10-s + 0.584i·11-s + (−0.227 + 0.0985i)12-s + (0.273 + 0.273i)13-s − 1.31·14-s + (−0.429 + 0.903i)15-s + 0.690·16-s + (0.0612 + 0.0612i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.542 - 0.840i$
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: \((2,\ 15,\ (\ :9/2),\ -0.542 - 0.840i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.299989 + 0.550831i\)
\(L(\frac12)\) \(\approx\) \(0.299989 + 0.550831i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (55.7 + 128. i)T \)
5 \( 1 + (919. + 1.05e3i)T \)
good2 \( 1 + (13.8 + 13.8i)T + 512iT^{2} \)
7 \( 1 + (-6.79e3 + 6.79e3i)T - 4.03e7iT^{2} \)
11 \( 1 - 2.84e4iT - 2.35e9T^{2} \)
13 \( 1 + (-2.82e4 - 2.82e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-2.11e4 - 2.11e4i)T + 1.18e11iT^{2} \)
19 \( 1 + 7.59e5iT - 3.22e11T^{2} \)
23 \( 1 + (-1.18e6 + 1.18e6i)T - 1.80e12iT^{2} \)
29 \( 1 + 6.93e6T + 1.45e13T^{2} \)
31 \( 1 + 1.45e6T + 2.64e13T^{2} \)
37 \( 1 + (8.18e6 - 8.18e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 7.44e6iT - 3.27e14T^{2} \)
43 \( 1 + (-6.21e6 - 6.21e6i)T + 5.02e14iT^{2} \)
47 \( 1 + (1.95e7 + 1.95e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (-4.55e7 + 4.55e7i)T - 3.29e15iT^{2} \)
59 \( 1 + 3.03e7T + 8.66e15T^{2} \)
61 \( 1 + 5.71e7T + 1.16e16T^{2} \)
67 \( 1 + (1.30e8 - 1.30e8i)T - 2.72e16iT^{2} \)
71 \( 1 - 1.62e8iT - 4.58e16T^{2} \)
73 \( 1 + (1.40e8 + 1.40e8i)T + 5.88e16iT^{2} \)
79 \( 1 + 4.65e7iT - 1.19e17T^{2} \)
83 \( 1 + (-4.75e8 + 4.75e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 1.99e8T + 3.50e17T^{2} \)
97 \( 1 + (-9.83e8 + 9.83e8i)T - 7.60e17iT^{2} \)
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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

−16.86585765544781778002466483172, −14.80455952441266451457854841308, −13.23071372956528345035929598737, −11.67459333661293378671272323946, −10.84384196094844916072076417386, −8.743892424343142593103010956866, −7.30343316163147304202935089385, −4.92081530713883273695410109915, −1.61294115460225820636341835544, −0.43936326237765545586960704685, 3.50104210256991059259868989765, 5.76171649528276590495038762609, 7.81461343333457643970084516504, 9.063156442074727768560495151489, 10.97283333777159452372153553178, 12.02735426810674997669702731505, 14.73511299492955509381455879968, 15.48206822704394011483981984969, 16.63040295489745813315559078215, 17.92959531725497011832271047026

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