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Properties

Label 2-15-5.2-c10-0-2
Degree $2$
Conductor $15$
Sign $0.661 + 0.749i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−28.0 − 28.0i)2-s + (99.2 − 99.2i)3-s + 554. i·4-s + (2.55e3 + 1.80e3i)5-s − 5.57e3·6-s + (2.01e4 + 2.01e4i)7-s + (−1.31e4 + 1.31e4i)8-s − 1.96e4i·9-s + (−2.09e4 − 1.22e5i)10-s + 1.09e5·11-s + (5.50e4 + 5.50e4i)12-s + (2.55e5 − 2.55e5i)13-s − 1.13e6i·14-s + (4.32e5 − 7.38e4i)15-s + 1.30e6·16-s + (−1.32e6 − 1.32e6i)17-s + ⋯
L(s)  = 1  + (−0.877 − 0.877i)2-s + (0.408 − 0.408i)3-s + 0.541i·4-s + (0.816 + 0.577i)5-s − 0.716·6-s + (1.19 + 1.19i)7-s + (−0.402 + 0.402i)8-s − 0.333i·9-s + (−0.209 − 1.22i)10-s + 0.678·11-s + (0.221 + 0.221i)12-s + (0.687 − 0.687i)13-s − 2.10i·14-s + (0.569 − 0.0972i)15-s + 1.24·16-s + (−0.934 − 0.934i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.661 + 0.749i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ 0.661 + 0.749i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.45579 - 0.657067i\)
\(L(\frac12)\) \(\approx\) \(1.45579 - 0.657067i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-99.2 + 99.2i)T \)
5 \( 1 + (-2.55e3 - 1.80e3i)T \)
good2 \( 1 + (28.0 + 28.0i)T + 1.02e3iT^{2} \)
7 \( 1 + (-2.01e4 - 2.01e4i)T + 2.82e8iT^{2} \)
11 \( 1 - 1.09e5T + 2.59e10T^{2} \)
13 \( 1 + (-2.55e5 + 2.55e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (1.32e6 + 1.32e6i)T + 2.01e12iT^{2} \)
19 \( 1 - 3.95e6iT - 6.13e12T^{2} \)
23 \( 1 + (-1.52e6 + 1.52e6i)T - 4.14e13iT^{2} \)
29 \( 1 + 1.53e7iT - 4.20e14T^{2} \)
31 \( 1 - 2.75e7T + 8.19e14T^{2} \)
37 \( 1 + (1.67e7 + 1.67e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 1.16e8T + 1.34e16T^{2} \)
43 \( 1 + (-7.69e7 + 7.69e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (1.29e8 + 1.29e8i)T + 5.25e16iT^{2} \)
53 \( 1 + (3.99e8 - 3.99e8i)T - 1.74e17iT^{2} \)
59 \( 1 - 1.93e8iT - 5.11e17T^{2} \)
61 \( 1 + 4.41e7T + 7.13e17T^{2} \)
67 \( 1 + (5.17e8 + 5.17e8i)T + 1.82e18iT^{2} \)
71 \( 1 + 1.06e9T + 3.25e18T^{2} \)
73 \( 1 + (-1.19e9 + 1.19e9i)T - 4.29e18iT^{2} \)
79 \( 1 - 1.89e9iT - 9.46e18T^{2} \)
83 \( 1 + (3.98e8 - 3.98e8i)T - 1.55e19iT^{2} \)
89 \( 1 - 3.69e9iT - 3.11e19T^{2} \)
97 \( 1 + (1.18e10 + 1.18e10i)T + 7.37e19iT^{2} \)
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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

−17.56875992490619745359912255030, −15.09031392074553354130081117601, −14.03768612784580500315226754783, −12.09522394518128708888976128911, −10.93483044943194957123842193462, −9.387079260311698285297091247998, −8.267661569866868833063894289508, −5.87969555777290264454952474182, −2.57403641370506010391669246189, −1.47194818106467675104949385838, 1.25352511218308021766231294261, 4.43243277364632726741643832706, 6.69830849488196716783471445582, 8.348130962106909960221799643655, 9.312161188715095334163470823036, 10.95152355704231665696810266118, 13.37095841148343914044544586269, 14.56064053165668952450225283451, 16.05885112907422375197693671580, 17.29202867501402592939728426249

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