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Properties

Label 2-15-15.8-c7-0-7
Degree $2$
Conductor $15$
Sign $0.573 + 0.819i$
Analytic cond. $4.68577$
Root an. cond. $2.16466$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.72 − 1.72i)2-s + (−28.9 + 36.7i)3-s − 122. i·4-s + (270. − 71.7i)5-s + (113. − 13.5i)6-s + (884. − 884. i)7-s + (−430. + 430. i)8-s + (−515. − 2.12e3i)9-s + (−588. − 341. i)10-s − 3.16e3i·11-s + (4.48e3 + 3.52e3i)12-s + (4.36e3 + 4.36e3i)13-s − 3.04e3·14-s + (−5.17e3 + 1.20e4i)15-s − 1.41e4·16-s + (−6.26e3 − 6.26e3i)17-s + ⋯
L(s)  = 1  + (−0.152 − 0.152i)2-s + (−0.618 + 0.786i)3-s − 0.953i·4-s + (0.966 − 0.256i)5-s + (0.213 − 0.0255i)6-s + (0.974 − 0.974i)7-s + (−0.297 + 0.297i)8-s + (−0.235 − 0.971i)9-s + (−0.186 − 0.107i)10-s − 0.716i·11-s + (0.749 + 0.589i)12-s + (0.551 + 0.551i)13-s − 0.296·14-s + (−0.395 + 0.918i)15-s − 0.863·16-s + (−0.309 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.573 + 0.819i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(2.16466\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :7/2),\ 0.573 + 0.819i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.21038 - 0.630370i\)
\(L(\frac12)\) \(\approx\) \(1.21038 - 0.630370i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (28.9 - 36.7i)T \)
5 \( 1 + (-270. + 71.7i)T \)
good2 \( 1 + (1.72 + 1.72i)T + 128iT^{2} \)
7 \( 1 + (-884. + 884. i)T - 8.23e5iT^{2} \)
11 \( 1 + 3.16e3iT - 1.94e7T^{2} \)
13 \( 1 + (-4.36e3 - 4.36e3i)T + 6.27e7iT^{2} \)
17 \( 1 + (6.26e3 + 6.26e3i)T + 4.10e8iT^{2} \)
19 \( 1 - 1.70e4iT - 8.93e8T^{2} \)
23 \( 1 + (-4.35e4 + 4.35e4i)T - 3.40e9iT^{2} \)
29 \( 1 + 2.22e5T + 1.72e10T^{2} \)
31 \( 1 - 1.46e5T + 2.75e10T^{2} \)
37 \( 1 + (2.34e5 - 2.34e5i)T - 9.49e10iT^{2} \)
41 \( 1 - 4.71e5iT - 1.94e11T^{2} \)
43 \( 1 + (-2.97e5 - 2.97e5i)T + 2.71e11iT^{2} \)
47 \( 1 + (-7.03e5 - 7.03e5i)T + 5.06e11iT^{2} \)
53 \( 1 + (-8.79e3 + 8.79e3i)T - 1.17e12iT^{2} \)
59 \( 1 - 3.79e4T + 2.48e12T^{2} \)
61 \( 1 + 2.50e5T + 3.14e12T^{2} \)
67 \( 1 + (1.78e6 - 1.78e6i)T - 6.06e12iT^{2} \)
71 \( 1 + 4.43e6iT - 9.09e12T^{2} \)
73 \( 1 + (-5.60e5 - 5.60e5i)T + 1.10e13iT^{2} \)
79 \( 1 + 4.39e6iT - 1.92e13T^{2} \)
83 \( 1 + (-2.63e5 + 2.63e5i)T - 2.71e13iT^{2} \)
89 \( 1 - 9.13e6T + 4.42e13T^{2} \)
97 \( 1 + (8.07e6 - 8.07e6i)T - 8.07e13iT^{2} \)
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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

−17.44001486455167476289945828781, −16.41881887185599732291782905307, −14.71691429567851781129574006413, −13.70774379032126116279428177665, −11.26549199697244742300036528970, −10.44106727793018151341826888608, −9.082876215667822520109515034855, −6.16388924640508461291702051853, −4.73584328781242521454079097221, −1.12741050999269940049277353239, 2.12290084029260747332070806581, 5.49739760747173298569159461154, 7.23365367252301634065960078520, 8.805977107186927430228649362241, 11.09116642561850952787344765607, 12.40729200741357365263005278797, 13.51070122762262569249381219303, 15.33385484596331017975603804667, 17.22494357710586069927444098623, 17.72211437792792931261386944072

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