Properties

Label 2-15-5.4-c9-0-4
Degree $2$
Conductor $15$
Sign $0.969 + 0.245i$
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  + 29.7i·2-s − 81i·3-s − 373.·4-s + (343. − 1.35e3i)5-s + 2.40e3·6-s − 1.07e4i·7-s + 4.13e3i·8-s − 6.56e3·9-s + (4.03e4 + 1.02e4i)10-s + 5.44e4·11-s + 3.02e4i·12-s + 5.35e4i·13-s + 3.18e5·14-s + (−1.09e5 − 2.78e4i)15-s − 3.13e5·16-s − 6.44e5i·17-s + ⋯
L(s)  = 1  + 1.31i·2-s − 0.577i·3-s − 0.728·4-s + (0.245 − 0.969i)5-s + 0.759·6-s − 1.68i·7-s + 0.356i·8-s − 0.333·9-s + (1.27 + 0.323i)10-s + 1.12·11-s + 0.420i·12-s + 0.519i·13-s + 2.21·14-s + (−0.559 − 0.141i)15-s − 1.19·16-s − 1.87i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.969 + 0.245i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ 0.969 + 0.245i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.67350 - 0.208833i\)
\(L(\frac12)\) \(\approx\) \(1.67350 - 0.208833i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81iT \)
5 \( 1 + (-343. + 1.35e3i)T \)
good2 \( 1 - 29.7iT - 512T^{2} \)
7 \( 1 + 1.07e4iT - 4.03e7T^{2} \)
11 \( 1 - 5.44e4T + 2.35e9T^{2} \)
13 \( 1 - 5.35e4iT - 1.06e10T^{2} \)
17 \( 1 + 6.44e5iT - 1.18e11T^{2} \)
19 \( 1 - 2.10e5T + 3.22e11T^{2} \)
23 \( 1 - 9.52e4iT - 1.80e12T^{2} \)
29 \( 1 - 2.25e6T + 1.45e13T^{2} \)
31 \( 1 + 5.48e5T + 2.64e13T^{2} \)
37 \( 1 - 1.19e7iT - 1.29e14T^{2} \)
41 \( 1 - 1.48e7T + 3.27e14T^{2} \)
43 \( 1 - 2.80e7iT - 5.02e14T^{2} \)
47 \( 1 + 5.85e6iT - 1.11e15T^{2} \)
53 \( 1 - 5.05e7iT - 3.29e15T^{2} \)
59 \( 1 - 5.84e6T + 8.66e15T^{2} \)
61 \( 1 + 1.07e7T + 1.16e16T^{2} \)
67 \( 1 - 7.46e7iT - 2.72e16T^{2} \)
71 \( 1 - 4.06e7T + 4.58e16T^{2} \)
73 \( 1 + 3.31e8iT - 5.88e16T^{2} \)
79 \( 1 - 8.40e7T + 1.19e17T^{2} \)
83 \( 1 - 6.88e8iT - 1.86e17T^{2} \)
89 \( 1 - 1.04e9T + 3.50e17T^{2} \)
97 \( 1 - 1.28e9iT - 7.60e17T^{2} \)
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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

−16.86235332210110757090495684642, −16.23017189538840069167717924047, −14.18084391123113425856583435825, −13.59008048512871170170068394800, −11.65232199251936106474116367116, −9.300652474075780121878158305361, −7.65697573113760645245414570357, −6.59052075004843036856490050916, −4.66401915695825710434697231049, −0.955486554575044734772895162382, 2.11630379973610380778822065997, 3.54232816679901321022532486156, 6.07471605351197105544272280434, 8.970064325986603370276539364579, 10.26238831714547390082896433568, 11.45393239954352505100904787589, 12.55435384820432006085869359678, 14.56382528058771620536915087587, 15.58306115817550856905887824665, 17.65943333752239688362778584499

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