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Properties

Label 2-15-5.3-c2-0-1
Degree 22
Conductor 1515
Sign 0.991+0.130i0.991 + 0.130i
Analytic cond. 0.4087200.408720
Root an. cond. 0.6393120.639312
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.224 − 0.224i)2-s + (−1.22 − 1.22i)3-s + 3.89i·4-s + (−4.67 − 1.77i)5-s − 0.550·6-s + (3.44 − 3.44i)7-s + (1.77 + 1.77i)8-s + 2.99i·9-s + (−1.44 + 0.651i)10-s + 11.3·11-s + (4.77 − 4.77i)12-s + (−5.55 − 5.55i)13-s − 1.55i·14-s + (3.55 + 7.89i)15-s − 14.7·16-s + (−17.3 + 17.3i)17-s + ⋯
L(s)  = 1  + (0.112 − 0.112i)2-s + (−0.408 − 0.408i)3-s + 0.974i·4-s + (−0.934 − 0.355i)5-s − 0.0917·6-s + (0.492 − 0.492i)7-s + (0.221 + 0.221i)8-s + 0.333i·9-s + (−0.144 + 0.0651i)10-s + 1.03·11-s + (0.397 − 0.397i)12-s + (−0.426 − 0.426i)13-s − 0.110i·14-s + (0.236 + 0.526i)15-s − 0.924·16-s + (−1.02 + 1.02i)17-s + ⋯

Functional equation

Λ(s)=(15s/2 ΓC(s) L(s)=((0.991+0.130i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(15s/2 ΓC(s+1) L(s)=((0.991+0.130i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1515    =    3⋅53 \cdot 5
Sign: 0.991+0.130i0.991 + 0.130i
Analytic conductor: 0.4087200.408720
Root analytic conductor: 0.6393120.639312
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ15(13,⋅)\chi_{15} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 15, ( :1), 0.991+0.130i)(2,\ 15,\ (\ :1),\ 0.991 + 0.130i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.712668−0.0467985i0.712668 - 0.0467985i
L(12)L(\frac12) ≈\approx 0.712668−0.0467985i0.712668 - 0.0467985i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(1.22+1.22i)T 1 + (1.22 + 1.22i)T
5 1+(4.67+1.77i)T 1 + (4.67 + 1.77i)T
good2 1+(−0.224+0.224i)T−4iT2 1 + (-0.224 + 0.224i)T - 4iT^{2}
7 1+(−3.44+3.44i)T−49iT2 1 + (-3.44 + 3.44i)T - 49iT^{2}
11 1−11.3T+121T2 1 - 11.3T + 121T^{2}
13 1+(5.55+5.55i)T+169iT2 1 + (5.55 + 5.55i)T + 169iT^{2}
17 1+(17.3−17.3i)T−289iT2 1 + (17.3 - 17.3i)T - 289iT^{2}
19 1+8.69iT−361T2 1 + 8.69iT - 361T^{2}
23 1+(−11.5−11.5i)T+529iT2 1 + (-11.5 - 11.5i)T + 529iT^{2}
29 1+35.1iT−841T2 1 + 35.1iT - 841T^{2}
31 1−10.6T+961T2 1 - 10.6T + 961T^{2}
37 1+(6.04−6.04i)T−1.36e3iT2 1 + (6.04 - 6.04i)T - 1.36e3iT^{2}
41 1−0.696T+1.68e3T2 1 - 0.696T + 1.68e3T^{2}
43 1+(26.4+26.4i)T+1.84e3iT2 1 + (26.4 + 26.4i)T + 1.84e3iT^{2}
47 1+(−44.2+44.2i)T−2.20e3iT2 1 + (-44.2 + 44.2i)T - 2.20e3iT^{2}
53 1+(0.696+0.696i)T+2.80e3iT2 1 + (0.696 + 0.696i)T + 2.80e3iT^{2}
59 1−39.9iT−3.48e3T2 1 - 39.9iT - 3.48e3T^{2}
61 1−5.90T+3.72e3T2 1 - 5.90T + 3.72e3T^{2}
67 1+(45.1−45.1i)T−4.48e3iT2 1 + (45.1 - 45.1i)T - 4.48e3iT^{2}
71 1+68T+5.04e3T2 1 + 68T + 5.04e3T^{2}
73 1+(−77.7−77.7i)T+5.32e3iT2 1 + (-77.7 - 77.7i)T + 5.32e3iT^{2}
79 1−24.4iT−6.24e3T2 1 - 24.4iT - 6.24e3T^{2}
83 1+(−13.1−13.1i)T+6.88e3iT2 1 + (-13.1 - 13.1i)T + 6.88e3iT^{2}
89 1+82.1iT−7.92e3T2 1 + 82.1iT - 7.92e3T^{2}
97 1+(24.5−24.5i)T−9.40e3iT2 1 + (24.5 - 24.5i)T - 9.40e3iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−19.38545099124929905717538849349, −17.50501961280439289905304112007, −16.91880099812900574531630198919, −15.31558856257637995376901438599, −13.40415428810468843990665511310, −12.15455676244450934468808544599, −11.19155019164708753230762264607, −8.502508879335912284238454206158, −7.17061287013729177032608512564, −4.23786525700189323868467818503, 4.70631905387741320263632042213, 6.73548236126271204075145647014, 9.138486862222952065413425180733, 10.89979022089326742614525957579, 11.91071577448152424986163173134, 14.30687367756745500554832403620, 15.16944591749910330032757698124, 16.34205360435462487295195706961, 18.10520000283866098205353209864, 19.24650811860761799651666020340

Graph of the ZZ-function along the critical line