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Properties

Label 2-15-15.14-c10-0-1
Degree $2$
Conductor $15$
Sign $0.158 - 0.987i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 26.2·2-s + (−17.3 − 242. i)3-s − 333.·4-s + (−3.04e3 − 714. i)5-s + (456. + 6.36e3i)6-s − 2.21e4i·7-s + 3.56e4·8-s + (−5.84e4 + 8.41e3i)9-s + (7.99e4 + 1.87e4i)10-s + 2.12e5i·11-s + (5.79e3 + 8.08e4i)12-s + 1.46e5i·13-s + 5.81e5i·14-s + (−1.20e5 + 7.49e5i)15-s − 5.95e5·16-s + 1.94e5·17-s + ⋯
L(s)  = 1  − 0.821·2-s + (−0.0714 − 0.997i)3-s − 0.325·4-s + (−0.973 − 0.228i)5-s + (0.0586 + 0.819i)6-s − 1.31i·7-s + 1.08·8-s + (−0.989 + 0.142i)9-s + (0.799 + 0.187i)10-s + 1.32i·11-s + (0.0232 + 0.324i)12-s + 0.395i·13-s + 1.08i·14-s + (−0.158 + 0.987i)15-s − 0.568·16-s + 0.137·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.158 - 0.987i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ 0.158 - 0.987i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.113463 + 0.0966902i\)
\(L(\frac12)\) \(\approx\) \(0.113463 + 0.0966902i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (17.3 + 242. i)T \)
5 \( 1 + (3.04e3 + 714. i)T \)
good2 \( 1 + 26.2T + 1.02e3T^{2} \)
7 \( 1 + 2.21e4iT - 2.82e8T^{2} \)
11 \( 1 - 2.12e5iT - 2.59e10T^{2} \)
13 \( 1 - 1.46e5iT - 1.37e11T^{2} \)
17 \( 1 - 1.94e5T + 2.01e12T^{2} \)
19 \( 1 - 2.72e6T + 6.13e12T^{2} \)
23 \( 1 + 8.45e6T + 4.14e13T^{2} \)
29 \( 1 + 2.69e7iT - 4.20e14T^{2} \)
31 \( 1 + 1.28e7T + 8.19e14T^{2} \)
37 \( 1 + 3.69e7iT - 4.80e15T^{2} \)
41 \( 1 - 1.87e8iT - 1.34e16T^{2} \)
43 \( 1 - 1.57e8iT - 2.16e16T^{2} \)
47 \( 1 + 1.71e8T + 5.25e16T^{2} \)
53 \( 1 + 9.61e7T + 1.74e17T^{2} \)
59 \( 1 - 6.71e8iT - 5.11e17T^{2} \)
61 \( 1 + 2.00e8T + 7.13e17T^{2} \)
67 \( 1 + 1.13e8iT - 1.82e18T^{2} \)
71 \( 1 - 1.53e9iT - 3.25e18T^{2} \)
73 \( 1 - 3.09e9iT - 4.29e18T^{2} \)
79 \( 1 + 4.13e9T + 9.46e18T^{2} \)
83 \( 1 + 7.78e8T + 1.55e19T^{2} \)
89 \( 1 + 7.12e9iT - 3.11e19T^{2} \)
97 \( 1 + 7.15e8iT - 7.37e19T^{2} \)
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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

−17.44415964795620156012683158059, −16.35851220822175726509848936629, −14.26708475250369449502715833733, −13.01592088500558163343798093612, −11.55123372979458220983069579453, −9.833114962325881716446709780550, −7.960836649925966698757203112366, −7.25116760361162019045329909113, −4.31671768394515082218404266075, −1.24139989303243864094593631116, 0.11185601087364476613701474443, 3.43255964130091824785662393035, 5.38253036353671565781592836644, 8.176937093867968603654474874900, 9.096369060049995339320978789938, 10.65230923288451153124476716910, 11.92669887013027039623505199855, 14.19229030448442683427438018149, 15.65652784553448630904831153021, 16.40679659605999792070884552718

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