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Properties

Label 2-15-15.2-c9-0-9
Degree $2$
Conductor $15$
Sign $0.993 - 0.110i$
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  + (20.3 − 20.3i)2-s + (65.1 + 124. i)3-s − 318. i·4-s + (840. + 1.11e3i)5-s + (3.86e3 + 1.20e3i)6-s + (1.02e3 + 1.02e3i)7-s + (3.93e3 + 3.93e3i)8-s + (−1.11e4 + 1.61e4i)9-s + (3.98e4 + 5.62e3i)10-s − 8.23e4i·11-s + (3.96e4 − 2.07e4i)12-s + (4.61e4 − 4.61e4i)13-s + 4.17e4·14-s + (−8.39e4 + 1.77e5i)15-s + 3.23e5·16-s + (−2.83e5 + 2.83e5i)17-s + ⋯
L(s)  = 1  + (0.900 − 0.900i)2-s + (0.464 + 0.885i)3-s − 0.622i·4-s + (0.601 + 0.798i)5-s + (1.21 + 0.379i)6-s + (0.161 + 0.161i)7-s + (0.339 + 0.339i)8-s + (−0.568 + 0.822i)9-s + (1.26 + 0.177i)10-s − 1.69i·11-s + (0.551 − 0.289i)12-s + (0.448 − 0.448i)13-s + 0.290·14-s + (−0.428 + 0.903i)15-s + 1.23·16-s + (−0.822 + 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.993 - 0.110i$
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: \((2,\ 15,\ (\ :9/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(5)\) \(\approx\) \(3.22861 + 0.179425i\)
\(L(\frac12)\) \(\approx\) \(3.22861 + 0.179425i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-65.1 - 124. i)T \)
5 \( 1 + (-840. - 1.11e3i)T \)
good2 \( 1 + (-20.3 + 20.3i)T - 512iT^{2} \)
7 \( 1 + (-1.02e3 - 1.02e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 8.23e4iT - 2.35e9T^{2} \)
13 \( 1 + (-4.61e4 + 4.61e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (2.83e5 - 2.83e5i)T - 1.18e11iT^{2} \)
19 \( 1 + 4.48e5iT - 3.22e11T^{2} \)
23 \( 1 + (5.77e5 + 5.77e5i)T + 1.80e12iT^{2} \)
29 \( 1 + 3.17e6T + 1.45e13T^{2} \)
31 \( 1 - 4.84e6T + 2.64e13T^{2} \)
37 \( 1 + (2.00e6 + 2.00e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 2.19e7iT - 3.27e14T^{2} \)
43 \( 1 + (-5.07e6 + 5.07e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (-1.80e7 + 1.80e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-2.88e7 - 2.88e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 1.79e8T + 8.66e15T^{2} \)
61 \( 1 + 2.05e7T + 1.16e16T^{2} \)
67 \( 1 + (-1.55e8 - 1.55e8i)T + 2.72e16iT^{2} \)
71 \( 1 - 3.49e8iT - 4.58e16T^{2} \)
73 \( 1 + (-5.01e7 + 5.01e7i)T - 5.88e16iT^{2} \)
79 \( 1 - 5.64e8iT - 1.19e17T^{2} \)
83 \( 1 + (2.73e8 + 2.73e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 2.28e8T + 3.50e17T^{2} \)
97 \( 1 + (-7.06e8 - 7.06e8i)T + 7.60e17iT^{2} \)
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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

−17.11620390787438839614098942980, −15.44537402061844453636738248206, −14.11012158871063111896577414607, −13.37081899019026076534167594643, −11.22955464693843613973080543153, −10.53660625422823914161679215961, −8.571426646246430399129378516553, −5.67530807976567552691975641882, −3.74234967453663335674523362810, −2.50628943717088991647049207434, 1.64562043147979521631570842841, 4.56858840435972551692288182717, 6.28505739584092701124864310381, 7.64200702043337632827307666218, 9.509586821696460061024912625900, 12.27525003631240103205179503995, 13.32363893999688761792152078618, 14.21919543220009430809677309686, 15.51428481909222764937760984687, 17.03655159719293437593623200370

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