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Properties

Label 2-15-15.8-c17-0-29
Degree $2$
Conductor $15$
Sign $-0.990 - 0.136i$
Analytic cond. $27.4833$
Root an. cond. $5.24245$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (132. + 132. i)2-s + (−8.46e3 − 7.57e3i)3-s − 9.62e4i·4-s + (−7.96e5 − 3.59e5i)5-s + (−1.17e5 − 2.11e6i)6-s + (1.63e7 − 1.63e7i)7-s + (3.00e7 − 3.00e7i)8-s + (1.43e7 + 1.28e8i)9-s + (−5.76e7 − 1.52e8i)10-s − 1.29e9i·11-s + (−7.28e8 + 8.14e8i)12-s + (6.39e8 + 6.39e8i)13-s + 4.30e9·14-s + (4.02e9 + 9.07e9i)15-s − 4.68e9·16-s + (−2.64e10 − 2.64e10i)17-s + ⋯
L(s)  = 1  + (0.364 + 0.364i)2-s + (−0.745 − 0.666i)3-s − 0.734i·4-s + (−0.911 − 0.411i)5-s + (−0.0286 − 0.514i)6-s + (1.06 − 1.06i)7-s + (0.632 − 0.632i)8-s + (0.110 + 0.993i)9-s + (−0.182 − 0.482i)10-s − 1.81i·11-s + (−0.489 + 0.547i)12-s + (0.217 + 0.217i)13-s + 0.780·14-s + (0.405 + 0.914i)15-s − 0.272·16-s + (−0.919 − 0.919i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.990 - 0.136i$
Analytic conductor: \(27.4833\)
Root analytic conductor: \(5.24245\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :17/2),\ -0.990 - 0.136i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.248732190\)
\(L(\frac12)\) \(\approx\) \(1.248732190\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.46e3 + 7.57e3i)T \)
5 \( 1 + (7.96e5 + 3.59e5i)T \)
good2 \( 1 + (-132. - 132. i)T + 1.31e5iT^{2} \)
7 \( 1 + (-1.63e7 + 1.63e7i)T - 2.32e14iT^{2} \)
11 \( 1 + 1.29e9iT - 5.05e17T^{2} \)
13 \( 1 + (-6.39e8 - 6.39e8i)T + 8.65e18iT^{2} \)
17 \( 1 + (2.64e10 + 2.64e10i)T + 8.27e20iT^{2} \)
19 \( 1 - 8.36e10iT - 5.48e21T^{2} \)
23 \( 1 + (-1.23e11 + 1.23e11i)T - 1.41e23iT^{2} \)
29 \( 1 + 1.79e12T + 7.25e24T^{2} \)
31 \( 1 - 1.01e12T + 2.25e25T^{2} \)
37 \( 1 + (1.26e13 - 1.26e13i)T - 4.56e26iT^{2} \)
41 \( 1 - 1.80e12iT - 2.61e27T^{2} \)
43 \( 1 + (-2.94e13 - 2.94e13i)T + 5.87e27iT^{2} \)
47 \( 1 + (-9.02e13 - 9.02e13i)T + 2.66e28iT^{2} \)
53 \( 1 + (-2.64e14 + 2.64e14i)T - 2.05e29iT^{2} \)
59 \( 1 - 2.05e15T + 1.27e30T^{2} \)
61 \( 1 + 8.15e14T + 2.24e30T^{2} \)
67 \( 1 + (-1.73e14 + 1.73e14i)T - 1.10e31iT^{2} \)
71 \( 1 - 3.34e15iT - 2.96e31T^{2} \)
73 \( 1 + (-5.46e15 - 5.46e15i)T + 4.74e31iT^{2} \)
79 \( 1 + 1.34e16iT - 1.81e32T^{2} \)
83 \( 1 + (1.67e16 - 1.67e16i)T - 4.21e32iT^{2} \)
89 \( 1 + 2.60e16T + 1.37e33T^{2} \)
97 \( 1 + (2.64e16 - 2.64e16i)T - 5.95e33iT^{2} \)
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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

−14.24625871946165776240604118774, −13.37267394607854575913723982537, −11.46561397677740332490138126325, −10.82350474594671372046036804925, −8.249991860576601007056467156874, −6.96489834562498951122601737907, −5.45702872359472225990791122400, −4.22765338112205889390563101070, −1.24383910181377215076341498836, −0.44486860438737865131256888967, 2.22971549526175064946496593019, 4.03526148386146545974142696595, 4.97380455770927899537910675957, 7.19374029234146970645151338964, 8.767334092343123086042747867833, 10.85704706483341181275566158806, 11.73875127491709648698006686201, 12.61098334763962647939130853665, 14.99643630794384524772291847092, 15.52023423801685642742665328736

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