Properties

Label 2-15-15.2-c9-0-8
Degree $2$
Conductor $15$
Sign $0.825 - 0.564i$
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  + (−30.5 + 30.5i)2-s + (−54.5 + 129. i)3-s − 1.35e3i·4-s + (1.26e3 − 602. i)5-s + (−2.28e3 − 5.60e3i)6-s + (−2.74e3 − 2.74e3i)7-s + (2.56e4 + 2.56e4i)8-s + (−1.37e4 − 1.40e4i)9-s + (−2.01e4 + 5.68e4i)10-s − 4.81e3i·11-s + (1.74e5 + 7.36e4i)12-s + (6.67e4 − 6.67e4i)13-s + 1.67e5·14-s + (9.12e3 + 1.95e5i)15-s − 8.71e5·16-s + (−3.46e4 + 3.46e4i)17-s + ⋯
L(s)  = 1  + (−1.34 + 1.34i)2-s + (−0.388 + 0.921i)3-s − 2.63i·4-s + (0.902 − 0.430i)5-s + (−0.718 − 1.76i)6-s + (−0.432 − 0.432i)7-s + (2.21 + 2.21i)8-s + (−0.698 − 0.715i)9-s + (−0.635 + 1.79i)10-s − 0.0991i·11-s + (2.43 + 1.02i)12-s + (0.648 − 0.648i)13-s + 1.16·14-s + (0.0465 + 0.998i)15-s − 3.32·16-s + (−0.100 + 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.825 - 0.564i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ 0.825 - 0.564i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.660484 + 0.204351i\)
\(L(\frac12)\) \(\approx\) \(0.660484 + 0.204351i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (54.5 - 129. i)T \)
5 \( 1 + (-1.26e3 + 602. i)T \)
good2 \( 1 + (30.5 - 30.5i)T - 512iT^{2} \)
7 \( 1 + (2.74e3 + 2.74e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 4.81e3iT - 2.35e9T^{2} \)
13 \( 1 + (-6.67e4 + 6.67e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (3.46e4 - 3.46e4i)T - 1.18e11iT^{2} \)
19 \( 1 + 4.27e5iT - 3.22e11T^{2} \)
23 \( 1 + (-6.70e5 - 6.70e5i)T + 1.80e12iT^{2} \)
29 \( 1 - 8.83e5T + 1.45e13T^{2} \)
31 \( 1 - 7.22e6T + 2.64e13T^{2} \)
37 \( 1 + (5.43e6 + 5.43e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 2.76e7iT - 3.27e14T^{2} \)
43 \( 1 + (1.38e7 - 1.38e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (-1.97e6 + 1.97e6i)T - 1.11e15iT^{2} \)
53 \( 1 + (1.15e7 + 1.15e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 2.00e7T + 8.66e15T^{2} \)
61 \( 1 - 1.28e8T + 1.16e16T^{2} \)
67 \( 1 + (9.09e7 + 9.09e7i)T + 2.72e16iT^{2} \)
71 \( 1 - 4.40e7iT - 4.58e16T^{2} \)
73 \( 1 + (-3.10e8 + 3.10e8i)T - 5.88e16iT^{2} \)
79 \( 1 + 3.03e8iT - 1.19e17T^{2} \)
83 \( 1 + (-4.46e8 - 4.46e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 1.06e9T + 3.50e17T^{2} \)
97 \( 1 + (7.40e8 + 7.40e8i)T + 7.60e17iT^{2} \)
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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

−17.21381307093299755559232677479, −16.21513143126234924483875014714, −15.28137708463497524795781559428, −13.73688006290114518731160806644, −10.69881690862611281820998266221, −9.723380383751360618901059090869, −8.626626003093648444428330815472, −6.51558948283215179864610366058, −5.26509243513918076873623245517, −0.66738072095768990348163283088, 1.33880598422250727983123895349, 2.68875623403361099318464737949, 6.63654124582845958602623574311, 8.475594512208862974520226932237, 9.917587916774334408280452768832, 11.22487352635968036838352102709, 12.42877507945084369702240260050, 13.60529399213196653121050474722, 16.57003231670163543272927242259, 17.60683268894353380547276512535

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