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Properties

Label 2-285-15.2-c1-0-16
Degree 22
Conductor 285285
Sign 0.374−0.927i0.374 - 0.927i
Analytic cond. 2.275732.27573
Root an. cond. 1.508551.50855
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.538 + 0.538i)2-s + (1.54 + 0.777i)3-s + 1.42i·4-s + (1.01 − 1.99i)5-s + (−1.25 + 0.415i)6-s + (1.88 + 1.88i)7-s + (−1.84 − 1.84i)8-s + (1.79 + 2.40i)9-s + (0.524 + 1.62i)10-s − 1.35i·11-s + (−1.10 + 2.19i)12-s + (1.03 − 1.03i)13-s − 2.02·14-s + (3.12 − 2.29i)15-s − 0.856·16-s + (−1.42 + 1.42i)17-s + ⋯
L(s)  = 1  + (−0.380 + 0.380i)2-s + (0.893 + 0.448i)3-s + 0.710i·4-s + (0.455 − 0.890i)5-s + (−0.511 + 0.169i)6-s + (0.711 + 0.711i)7-s + (−0.651 − 0.651i)8-s + (0.597 + 0.801i)9-s + (0.165 + 0.512i)10-s − 0.409i·11-s + (−0.318 + 0.634i)12-s + (0.288 − 0.288i)13-s − 0.541·14-s + (0.806 − 0.591i)15-s − 0.214·16-s + (−0.345 + 0.345i)17-s + ⋯

Functional equation

Λ(s)=(285s/2 ΓC(s) L(s)=((0.374−0.927i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(285s/2 ΓC(s+1/2) L(s)=((0.374−0.927i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 285285    =    3⋅5⋅193 \cdot 5 \cdot 19
Sign: 0.374−0.927i0.374 - 0.927i
Analytic conductor: 2.275732.27573
Root analytic conductor: 1.508551.50855
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ285(77,⋅)\chi_{285} (77, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 285, ( :1/2), 0.374−0.927i)(2,\ 285,\ (\ :1/2),\ 0.374 - 0.927i)

Particular Values

L(1)L(1) ≈\approx 1.27824+0.861901i1.27824 + 0.861901i
L(12)L(\frac12) ≈\approx 1.27824+0.861901i1.27824 + 0.861901i
L(32)L(\frac{3}{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+(−1.54−0.777i)T 1 + (-1.54 - 0.777i)T
5 1+(−1.01+1.99i)T 1 + (-1.01 + 1.99i)T
19 1−iT 1 - iT
good2 1+(0.538−0.538i)T−2iT2 1 + (0.538 - 0.538i)T - 2iT^{2}
7 1+(−1.88−1.88i)T+7iT2 1 + (-1.88 - 1.88i)T + 7iT^{2}
11 1+1.35iT−11T2 1 + 1.35iT - 11T^{2}
13 1+(−1.03+1.03i)T−13iT2 1 + (-1.03 + 1.03i)T - 13iT^{2}
17 1+(1.42−1.42i)T−17iT2 1 + (1.42 - 1.42i)T - 17iT^{2}
23 1+(1.15+1.15i)T+23iT2 1 + (1.15 + 1.15i)T + 23iT^{2}
29 1+4.57T+29T2 1 + 4.57T + 29T^{2}
31 1+4.81T+31T2 1 + 4.81T + 31T^{2}
37 1+(−2.99−2.99i)T+37iT2 1 + (-2.99 - 2.99i)T + 37iT^{2}
41 1−8.00iT−41T2 1 - 8.00iT - 41T^{2}
43 1+(−5.08+5.08i)T−43iT2 1 + (-5.08 + 5.08i)T - 43iT^{2}
47 1+(−6.64+6.64i)T−47iT2 1 + (-6.64 + 6.64i)T - 47iT^{2}
53 1+(7.34+7.34i)T+53iT2 1 + (7.34 + 7.34i)T + 53iT^{2}
59 1−1.44T+59T2 1 - 1.44T + 59T^{2}
61 1+8.86T+61T2 1 + 8.86T + 61T^{2}
67 1+(2.63+2.63i)T+67iT2 1 + (2.63 + 2.63i)T + 67iT^{2}
71 1+4.15iT−71T2 1 + 4.15iT - 71T^{2}
73 1+(−10.4+10.4i)T−73iT2 1 + (-10.4 + 10.4i)T - 73iT^{2}
79 1+13.1iT−79T2 1 + 13.1iT - 79T^{2}
83 1+(−5.76−5.76i)T+83iT2 1 + (-5.76 - 5.76i)T + 83iT^{2}
89 1+15.3T+89T2 1 + 15.3T + 89T^{2}
97 1+(6.91+6.91i)T+97iT2 1 + (6.91 + 6.91i)T + 97iT^{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

−12.17024341124234980911405734895, −10.99747506974957067550931015058, −9.682995738073443043149087968191, −8.882556972003897326547524604009, −8.387077463683960989546206244850, −7.61948102909775845288940230940, −6.01530958426357082149786903960, −4.79147048293473680247101226384, −3.57923093320945749115951098759, −2.08190425512352371396647719257, 1.52716818507575925836757234439, 2.58212441661522093378193821347, 4.16085227855309817224790577894, 5.79231246348399298672219063086, 6.96027839264144736751780459639, 7.71550939187651938970260857011, 9.087857201070848731652990423645, 9.656306243621018949863133491750, 10.77016945758666160330913454522, 11.24442034876770159619300722573

Graph of the ZZ-function along the critical line