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Properties

Label 2-15-15.8-c9-0-8
Degree $2$
Conductor $15$
Sign $0.970 + 0.241i$
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 + (−128. − 55.7i)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 + (−7.07e3 + 1.63e4i)12-s + (2.82e4 + 2.82e4i)13-s + 1.88e5·14-s + (−5.98e4 − 1.86e5i)15-s + 1.80e5·16-s + (−2.11e4 − 2.11e4i)17-s + ⋯
L(s)  = 1  + (0.613 + 0.613i)2-s + (−0.917 − 0.397i)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 + (−0.0985 + 0.227i)12-s + (0.273 + 0.273i)13-s + 1.31·14-s + (−0.305 − 0.952i)15-s + 0.690·16-s + (−0.0612 − 0.0612i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.970 + 0.241i$
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: \((2,\ 15,\ (\ :9/2),\ 0.970 + 0.241i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.10223 - 0.257165i\)
\(L(\frac12)\) \(\approx\) \(2.10223 - 0.257165i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (128. + 55.7i)T \)
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

−17.18350115970084991264546519350, −15.73149723031189242402157330665, −14.08214788647122562189204404101, −13.54729446141712830285842287757, −11.27550770119864913061143294639, −10.33986364248907556730168931469, −7.34057316718461635453748729917, −6.18779012550941920564825865321, −4.71004084941154078260094732568, −1.24256083628811256113388763175, 1.83567661626376736058409700254, 4.48979512228000860729620500285, 5.63645994820426355632499007844, 8.448660553751480692692102402552, 10.38243605927772581841800148433, 11.97602887861920986206907422532, 12.50257744676345532646495101934, 14.31144400575479433663968118474, 16.01821382232356246464114941272, 17.33167880624206195314410007155

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