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Properties

Label 2-15-5.3-c10-0-0
Degree $2$
Conductor $15$
Sign $-0.941 + 0.337i$
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  + (−12.5 + 12.5i)2-s + (−99.2 − 99.2i)3-s + 708. i·4-s + (3.10e3 − 350. i)5-s + 2.49e3·6-s + (−8.58e3 + 8.58e3i)7-s + (−2.17e4 − 2.17e4i)8-s + 1.96e4i·9-s + (−3.45e4 + 4.33e4i)10-s − 1.17e5·11-s + (7.03e4 − 7.03e4i)12-s + (−4.68e5 − 4.68e5i)13-s − 2.15e5i·14-s + (−3.42e5 − 2.73e5i)15-s − 1.79e5·16-s + (−1.23e6 + 1.23e6i)17-s + ⋯
L(s)  = 1  + (−0.392 + 0.392i)2-s + (−0.408 − 0.408i)3-s + 0.692i·4-s + (0.993 − 0.112i)5-s + 0.320·6-s + (−0.510 + 0.510i)7-s + (−0.663 − 0.663i)8-s + 0.333i·9-s + (−0.345 + 0.433i)10-s − 0.728·11-s + (0.282 − 0.282i)12-s + (−1.26 − 1.26i)13-s − 0.400i·14-s + (−0.451 − 0.359i)15-s − 0.171·16-s + (−0.871 + 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.941 + 0.337i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ -0.941 + 0.337i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0272938 - 0.156967i\)
\(L(\frac12)\) \(\approx\) \(0.0272938 - 0.156967i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (99.2 + 99.2i)T \)
5 \( 1 + (-3.10e3 + 350. i)T \)
good2 \( 1 + (12.5 - 12.5i)T - 1.02e3iT^{2} \)
7 \( 1 + (8.58e3 - 8.58e3i)T - 2.82e8iT^{2} \)
11 \( 1 + 1.17e5T + 2.59e10T^{2} \)
13 \( 1 + (4.68e5 + 4.68e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (1.23e6 - 1.23e6i)T - 2.01e12iT^{2} \)
19 \( 1 + 7.33e5iT - 6.13e12T^{2} \)
23 \( 1 + (1.35e6 + 1.35e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 2.76e7iT - 4.20e14T^{2} \)
31 \( 1 + 2.91e7T + 8.19e14T^{2} \)
37 \( 1 + (-2.07e7 + 2.07e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 5.61e7T + 1.34e16T^{2} \)
43 \( 1 + (1.82e8 + 1.82e8i)T + 2.16e16iT^{2} \)
47 \( 1 + (-5.88e7 + 5.88e7i)T - 5.25e16iT^{2} \)
53 \( 1 + (-4.10e8 - 4.10e8i)T + 1.74e17iT^{2} \)
59 \( 1 - 1.10e9iT - 5.11e17T^{2} \)
61 \( 1 - 1.20e9T + 7.13e17T^{2} \)
67 \( 1 + (8.20e8 - 8.20e8i)T - 1.82e18iT^{2} \)
71 \( 1 + 4.17e8T + 3.25e18T^{2} \)
73 \( 1 + (-2.51e8 - 2.51e8i)T + 4.29e18iT^{2} \)
79 \( 1 - 4.16e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.13e9 + 1.13e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 3.39e9iT - 3.11e19T^{2} \)
97 \( 1 + (-4.57e9 + 4.57e9i)T - 7.37e19iT^{2} \)
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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

−17.62569844588917055693752982593, −16.64405624714310705304491045611, −15.22037708084269108456169943744, −13.11129596868363961624469126272, −12.47863305357741269625305016274, −10.30901661283166969857108652427, −8.730707811789812236225170842452, −7.08986713502174664476676883611, −5.53984139962305278722012207181, −2.58038479811997243172187573430, 0.083058773195178567646451488743, 2.19238232973695469871345999987, 5.02385155816813117948688703368, 6.61731229822483509184221499229, 9.448089933345717906070539821553, 10.09468834197877572032669868814, 11.49551993929119858616499624931, 13.46954670213990665355844483013, 14.73894305165306362950936845746, 16.37314139343984277753746719334

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