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Properties

Label 2-15-15.14-c8-0-0
Degree $2$
Conductor $15$
Sign $-0.979 + 0.201i$
Analytic cond. $6.11067$
Root an. cond. $2.47197$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 29.2·2-s + (−12.3 + 80.0i)3-s + 596.·4-s + (218. + 585. i)5-s + (360. − 2.33e3i)6-s + 2.07e3i·7-s − 9.94e3·8-s + (−6.25e3 − 1.97e3i)9-s + (−6.36e3 − 1.71e4i)10-s + 1.19e4i·11-s + (−7.37e3 + 4.77e4i)12-s − 2.78e4i·13-s − 6.05e4i·14-s + (−4.95e4 + 1.02e4i)15-s + 1.37e5·16-s + 3.24e4·17-s + ⋯
L(s)  = 1  − 1.82·2-s + (−0.152 + 0.988i)3-s + 2.33·4-s + (0.348 + 0.937i)5-s + (0.278 − 1.80i)6-s + 0.863i·7-s − 2.42·8-s + (−0.953 − 0.301i)9-s + (−0.636 − 1.71i)10-s + 0.817i·11-s + (−0.355 + 2.30i)12-s − 0.973i·13-s − 1.57i·14-s + (−0.979 + 0.201i)15-s + 2.10·16-s + 0.388·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.979 + 0.201i$
Analytic conductor: \(6.11067\)
Root analytic conductor: \(2.47197\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :4),\ -0.979 + 0.201i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0406350 - 0.398606i\)
\(L(\frac12)\) \(\approx\) \(0.0406350 - 0.398606i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (12.3 - 80.0i)T \)
5 \( 1 + (-218. - 585. i)T \)
good2 \( 1 + 29.2T + 256T^{2} \)
7 \( 1 - 2.07e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.19e4iT - 2.14e8T^{2} \)
13 \( 1 + 2.78e4iT - 8.15e8T^{2} \)
17 \( 1 - 3.24e4T + 6.97e9T^{2} \)
19 \( 1 + 1.22e5T + 1.69e10T^{2} \)
23 \( 1 + 2.66e5T + 7.83e10T^{2} \)
29 \( 1 + 7.93e5iT - 5.00e11T^{2} \)
31 \( 1 - 8.76e5T + 8.52e11T^{2} \)
37 \( 1 - 1.82e6iT - 3.51e12T^{2} \)
41 \( 1 - 1.49e5iT - 7.98e12T^{2} \)
43 \( 1 - 1.65e6iT - 1.16e13T^{2} \)
47 \( 1 + 3.35e6T + 2.38e13T^{2} \)
53 \( 1 + 2.01e6T + 6.22e13T^{2} \)
59 \( 1 + 2.16e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.66e7T + 1.91e14T^{2} \)
67 \( 1 - 2.51e7iT - 4.06e14T^{2} \)
71 \( 1 - 4.99e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.82e6iT - 8.06e14T^{2} \)
79 \( 1 - 2.56e7T + 1.51e15T^{2} \)
83 \( 1 - 4.83e7T + 2.25e15T^{2} \)
89 \( 1 + 3.21e6iT - 3.93e15T^{2} \)
97 \( 1 + 5.09e7iT - 7.83e15T^{2} \)
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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

−17.97478621992459504421240835691, −17.20512655981062214698851889789, −15.64828792962153905358767665709, −14.95238334804752129337547984656, −11.76820534882956800571532793820, −10.40719176284029061513402874778, −9.698611967155223770662018260416, −8.148627759312114298519053806525, −6.17499513643865952733518191214, −2.54071498532863518209098414562, 0.40825459513343473632458716615, 1.69434929780534584564754229665, 6.40671583150349281524868005771, 7.906556743306077312113076144929, 9.026829641610053044258359963368, 10.70398656661893992803894700923, 12.12640107659565713853375826924, 13.83536696680889548707137461977, 16.40092160532017209959215656160, 16.92472026543066666767015040554

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