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Properties

Label 2-15-15.2-c17-0-17
Degree $2$
Conductor $15$
Sign $0.228 + 0.973i$
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  + (−230. + 230. i)2-s + (−1.08e4 − 3.46e3i)3-s + 2.48e4i·4-s + (8.73e5 + 1.09e4i)5-s + (3.29e6 − 1.69e6i)6-s + (−7.64e6 − 7.64e6i)7-s + (−3.59e7 − 3.59e7i)8-s + (1.05e8 + 7.50e7i)9-s + (−2.03e8 + 1.98e8i)10-s + 1.20e9i·11-s + (8.63e7 − 2.69e8i)12-s + (−1.87e9 + 1.87e9i)13-s + 3.52e9·14-s + (−9.41e9 − 3.14e9i)15-s + 1.32e10·16-s + (−2.86e10 + 2.86e10i)17-s + ⋯
L(s)  = 1  + (−0.636 + 0.636i)2-s + (−0.952 − 0.305i)3-s + 0.189i·4-s + (0.999 + 0.0125i)5-s + (0.800 − 0.411i)6-s + (−0.500 − 0.500i)7-s + (−0.757 − 0.757i)8-s + (0.813 + 0.581i)9-s + (−0.644 + 0.628i)10-s + 1.69i·11-s + (0.0579 − 0.180i)12-s + (−0.636 + 0.636i)13-s + 0.637·14-s + (−0.948 − 0.317i)15-s + 0.774·16-s + (−0.997 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.228 + 0.973i$
Analytic conductor: \(27.4833\)
Root analytic conductor: \(5.24245\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :17/2),\ 0.228 + 0.973i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.2466609496\)
\(L(\frac12)\) \(\approx\) \(0.2466609496\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08e4 + 3.46e3i)T \)
5 \( 1 + (-8.73e5 - 1.09e4i)T \)
good2 \( 1 + (230. - 230. i)T - 1.31e5iT^{2} \)
7 \( 1 + (7.64e6 + 7.64e6i)T + 2.32e14iT^{2} \)
11 \( 1 - 1.20e9iT - 5.05e17T^{2} \)
13 \( 1 + (1.87e9 - 1.87e9i)T - 8.65e18iT^{2} \)
17 \( 1 + (2.86e10 - 2.86e10i)T - 8.27e20iT^{2} \)
19 \( 1 + 1.04e11iT - 5.48e21T^{2} \)
23 \( 1 + (1.97e10 + 1.97e10i)T + 1.41e23iT^{2} \)
29 \( 1 - 3.28e12T + 7.25e24T^{2} \)
31 \( 1 + 3.91e12T + 2.25e25T^{2} \)
37 \( 1 + (1.25e13 + 1.25e13i)T + 4.56e26iT^{2} \)
41 \( 1 - 1.41e13iT - 2.61e27T^{2} \)
43 \( 1 + (3.51e13 - 3.51e13i)T - 5.87e27iT^{2} \)
47 \( 1 + (-2.10e14 + 2.10e14i)T - 2.66e28iT^{2} \)
53 \( 1 + (2.40e14 + 2.40e14i)T + 2.05e29iT^{2} \)
59 \( 1 + 5.00e14T + 1.27e30T^{2} \)
61 \( 1 + 6.30e14T + 2.24e30T^{2} \)
67 \( 1 + (2.70e15 + 2.70e15i)T + 1.10e31iT^{2} \)
71 \( 1 + 3.65e15iT - 2.96e31T^{2} \)
73 \( 1 + (-4.07e15 + 4.07e15i)T - 4.74e31iT^{2} \)
79 \( 1 + 2.13e16iT - 1.81e32T^{2} \)
83 \( 1 + (-4.23e15 - 4.23e15i)T + 4.21e32iT^{2} \)
89 \( 1 + 2.76e16T + 1.37e33T^{2} \)
97 \( 1 + (7.24e15 + 7.24e15i)T + 5.95e33iT^{2} \)
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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

−15.29592788093868130737684047505, −13.26165575751088153209489248441, −12.31981141655085612056417056857, −10.37539554087115663223194474698, −9.257915891533689806064572806862, −7.12701365598769759618834487024, −6.54702918197684086846531499563, −4.63095963338267466073410810095, −1.99775340128942624610291580816, −0.12668878531039232680969466289, 1.08426447581533036024635314473, 2.79253142804491829318702188473, 5.39662092883985709415467636100, 6.19617103422981070016263030896, 8.938474088430027907228261343500, 10.03162084117977019256170568965, 10.98877675173992608585239970997, 12.33377247634463122205969161038, 14.01757097534903756885903456211, 15.76533069496927182710507803657

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