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Properties

Label 2-15-15.8-c11-0-16
Degree $2$
Conductor $15$
Sign $-0.991 - 0.133i$
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  + (−57.5 − 57.5i)2-s + (345. − 239. i)3-s + 4.57e3i·4-s + (1.29e3 − 6.86e3i)5-s + (−3.37e4 − 6.10e3i)6-s + (5.37e4 − 5.37e4i)7-s + (1.45e5 − 1.45e5i)8-s + (6.21e4 − 1.65e5i)9-s + (−4.69e5 + 3.20e5i)10-s − 2.99e5i·11-s + (1.09e6 + 1.58e6i)12-s + (2.57e5 + 2.57e5i)13-s − 6.19e6·14-s + (−1.19e6 − 2.68e6i)15-s − 7.38e6·16-s + (1.91e6 + 1.91e6i)17-s + ⋯
L(s)  = 1  + (−1.27 − 1.27i)2-s + (0.821 − 0.569i)3-s + 2.23i·4-s + (0.185 − 0.982i)5-s + (−1.76 − 0.320i)6-s + (1.20 − 1.20i)7-s + (1.57 − 1.57i)8-s + (0.350 − 0.936i)9-s + (−1.48 + 1.01i)10-s − 0.559i·11-s + (1.27 + 1.83i)12-s + (0.192 + 0.192i)13-s − 3.07·14-s + (−0.407 − 0.913i)15-s − 1.76·16-s + (0.326 + 0.326i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.133i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.991 - 0.133i$
Analytic conductor: \(11.5251\)
Root analytic conductor: \(3.39487\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :11/2),\ -0.991 - 0.133i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0900109 + 1.34321i\)
\(L(\frac12)\) \(\approx\) \(0.0900109 + 1.34321i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-345. + 239. i)T \)
5 \( 1 + (-1.29e3 + 6.86e3i)T \)
good2 \( 1 + (57.5 + 57.5i)T + 2.04e3iT^{2} \)
7 \( 1 + (-5.37e4 + 5.37e4i)T - 1.97e9iT^{2} \)
11 \( 1 + 2.99e5iT - 2.85e11T^{2} \)
13 \( 1 + (-2.57e5 - 2.57e5i)T + 1.79e12iT^{2} \)
17 \( 1 + (-1.91e6 - 1.91e6i)T + 3.42e13iT^{2} \)
19 \( 1 - 1.22e7iT - 1.16e14T^{2} \)
23 \( 1 + (1.69e4 - 1.69e4i)T - 9.52e14iT^{2} \)
29 \( 1 + 8.47e7T + 1.22e16T^{2} \)
31 \( 1 - 5.68e7T + 2.54e16T^{2} \)
37 \( 1 + (2.28e8 - 2.28e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 4.14e8iT - 5.50e17T^{2} \)
43 \( 1 + (-5.52e8 - 5.52e8i)T + 9.29e17iT^{2} \)
47 \( 1 + (3.42e8 + 3.42e8i)T + 2.47e18iT^{2} \)
53 \( 1 + (1.12e9 - 1.12e9i)T - 9.26e18iT^{2} \)
59 \( 1 - 6.76e9T + 3.01e19T^{2} \)
61 \( 1 - 3.58e9T + 4.35e19T^{2} \)
67 \( 1 + (-1.15e10 + 1.15e10i)T - 1.22e20iT^{2} \)
71 \( 1 - 2.04e10iT - 2.31e20T^{2} \)
73 \( 1 + (-8.20e9 - 8.20e9i)T + 3.13e20iT^{2} \)
79 \( 1 + 2.87e10iT - 7.47e20T^{2} \)
83 \( 1 + (-6.21e9 + 6.21e9i)T - 1.28e21iT^{2} \)
89 \( 1 + 9.02e10T + 2.77e21T^{2} \)
97 \( 1 + (1.89e9 - 1.89e9i)T - 7.15e21iT^{2} \)
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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

−16.68295006143306902824034867907, −14.09375069806829539373046164486, −12.84068222932641776883969853547, −11.49172101741255154557564055695, −9.988114125351698952029006187073, −8.529120651480692184224105492310, −7.79794981072640193159666336064, −3.84974988481447326945175267891, −1.70480614158187355739427377188, −0.911130249675211440011004195566, 2.15213274002663917646528975005, 5.34342973745611711821403780464, 7.27571197865634262851623294146, 8.487056349241221523803358167629, 9.600728389127467181555427833694, 11.00805341725560183948082133941, 14.21625149037990135473107089299, 15.05090900667800687303253391311, 15.68472481273056047802629604068, 17.51349770514606303856901719350

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