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Properties

Label 2-15-3.2-c8-0-9
Degree $2$
Conductor $15$
Sign $0.124 - 0.992i$
Analytic cond. $6.11067$
Root an. cond. $2.47197$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 23.7i·2-s + (−80.3 − 10.0i)3-s − 309.·4-s + 279. i·5-s + (−239. + 1.91e3i)6-s − 692.·7-s + 1.28e3i·8-s + (6.35e3 + 1.61e3i)9-s + 6.64e3·10-s + 1.58e4i·11-s + (2.49e4 + 3.12e3i)12-s − 4.88e4·13-s + 1.64e4i·14-s + (2.81e3 − 2.24e4i)15-s − 4.88e4·16-s + 4.10e4i·17-s + ⋯
L(s)  = 1  − 1.48i·2-s + (−0.992 − 0.124i)3-s − 1.21·4-s + 0.447i·5-s + (−0.184 + 1.47i)6-s − 0.288·7-s + 0.313i·8-s + (0.969 + 0.246i)9-s + 0.664·10-s + 1.08i·11-s + (1.20 + 0.150i)12-s − 1.71·13-s + 0.428i·14-s + (0.0556 − 0.443i)15-s − 0.745·16-s + 0.491i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.124 - 0.992i$
Analytic conductor: \(6.11067\)
Root analytic conductor: \(2.47197\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :4),\ 0.124 - 0.992i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.00212921 + 0.00187890i\)
\(L(\frac12)\) \(\approx\) \(0.00212921 + 0.00187890i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (80.3 + 10.0i)T \)
5 \( 1 - 279. iT \)
good2 \( 1 + 23.7iT - 256T^{2} \)
7 \( 1 + 692.T + 5.76e6T^{2} \)
11 \( 1 - 1.58e4iT - 2.14e8T^{2} \)
13 \( 1 + 4.88e4T + 8.15e8T^{2} \)
17 \( 1 - 4.10e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.08e5T + 1.69e10T^{2} \)
23 \( 1 + 4.33e5iT - 7.83e10T^{2} \)
29 \( 1 - 3.77e5iT - 5.00e11T^{2} \)
31 \( 1 + 4.08e5T + 8.52e11T^{2} \)
37 \( 1 - 2.31e6T + 3.51e12T^{2} \)
41 \( 1 + 2.15e6iT - 7.98e12T^{2} \)
43 \( 1 + 2.42e6T + 1.16e13T^{2} \)
47 \( 1 + 8.18e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.36e7iT - 6.22e13T^{2} \)
59 \( 1 + 1.23e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.15e6T + 1.91e14T^{2} \)
67 \( 1 + 1.55e7T + 4.06e14T^{2} \)
71 \( 1 - 2.52e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.29e7T + 8.06e14T^{2} \)
79 \( 1 - 4.49e7T + 1.51e15T^{2} \)
83 \( 1 - 2.12e7iT - 2.25e15T^{2} \)
89 \( 1 - 4.87e7iT - 3.93e15T^{2} \)
97 \( 1 - 2.91e7T + 7.83e15T^{2} \)
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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

−16.84096299381142244423420189327, −14.91847270642692345664143480875, −12.78109110578992603875082684633, −12.15004055175333892371105730895, −10.68438893344798807996891218592, −9.833995006209655293221016914370, −6.90545756323120609963415423390, −4.52138412402494469245117896135, −2.18993078904505299762957034694, −0.00174707290569415816943334302, 4.88519810827731728334358505837, 6.13351214780762367692668099154, 7.62032084354527081277694605945, 9.514614997250405458375904777368, 11.54592153544987913096992171883, 13.19569835729611611198912070891, 14.85441315258683616028604415398, 16.12457138491912406438096819353, 16.84462230105661449230381454191, 17.76878541796488859638917517101

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