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Properties

Label 2-360-120.53-c1-0-8
Degree 22
Conductor 360360
Sign 0.962−0.271i0.962 - 0.271i
Analytic cond. 2.874612.87461
Root an. cond. 1.695461.69546
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  + (−0.760 + 1.19i)2-s + (−0.844 − 1.81i)4-s + (−2.23 + 0.0113i)5-s + (0.471 − 0.471i)7-s + (2.80 + 0.372i)8-s + (1.68 − 2.67i)10-s + 0.335·11-s + (3.50 − 3.50i)13-s + (0.203 + 0.921i)14-s + (−2.57 + 3.06i)16-s + (2.53 + 2.53i)17-s + 4.07·19-s + (1.90 + 4.04i)20-s + (−0.255 + 0.400i)22-s + (6.20 − 6.20i)23-s + ⋯
L(s)  = 1  + (−0.537 + 0.843i)2-s + (−0.422 − 0.906i)4-s + (−0.999 + 0.00506i)5-s + (0.178 − 0.178i)7-s + (0.991 + 0.131i)8-s + (0.533 − 0.845i)10-s + 0.101·11-s + (0.971 − 0.971i)13-s + (0.0545 + 0.246i)14-s + (−0.643 + 0.765i)16-s + (0.614 + 0.614i)17-s + 0.934·19-s + (0.426 + 0.904i)20-s + (−0.0544 + 0.0853i)22-s + (1.29 − 1.29i)23-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((0.962−0.271i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+1/2) L(s)=((0.962−0.271i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: 0.962−0.271i0.962 - 0.271i
Analytic conductor: 2.874612.87461
Root analytic conductor: 1.695461.69546
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ360(53,⋅)\chi_{360} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :1/2), 0.962−0.271i)(2,\ 360,\ (\ :1/2),\ 0.962 - 0.271i)

Particular Values

L(1)L(1) ≈\approx 0.877354+0.121350i0.877354 + 0.121350i
L(12)L(\frac12) ≈\approx 0.877354+0.121350i0.877354 + 0.121350i
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)
bad2 1+(0.760−1.19i)T 1 + (0.760 - 1.19i)T
3 1 1
5 1+(2.23−0.0113i)T 1 + (2.23 - 0.0113i)T
good7 1+(−0.471+0.471i)T−7iT2 1 + (-0.471 + 0.471i)T - 7iT^{2}
11 1−0.335T+11T2 1 - 0.335T + 11T^{2}
13 1+(−3.50+3.50i)T−13iT2 1 + (-3.50 + 3.50i)T - 13iT^{2}
17 1+(−2.53−2.53i)T+17iT2 1 + (-2.53 - 2.53i)T + 17iT^{2}
19 1−4.07T+19T2 1 - 4.07T + 19T^{2}
23 1+(−6.20+6.20i)T−23iT2 1 + (-6.20 + 6.20i)T - 23iT^{2}
29 1+2.42iT−29T2 1 + 2.42iT - 29T^{2}
31 1+6.41T+31T2 1 + 6.41T + 31T^{2}
37 1+(−2.24−2.24i)T+37iT2 1 + (-2.24 - 2.24i)T + 37iT^{2}
41 1−5.80iT−41T2 1 - 5.80iT - 41T^{2}
43 1+(−4.87+4.87i)T−43iT2 1 + (-4.87 + 4.87i)T - 43iT^{2}
47 1+(1.68+1.68i)T+47iT2 1 + (1.68 + 1.68i)T + 47iT^{2}
53 1+(3.05+3.05i)T+53iT2 1 + (3.05 + 3.05i)T + 53iT^{2}
59 1+12.2iT−59T2 1 + 12.2iT - 59T^{2}
61 1−7.49iT−61T2 1 - 7.49iT - 61T^{2}
67 1+(5.55+5.55i)T+67iT2 1 + (5.55 + 5.55i)T + 67iT^{2}
71 1+13.4iT−71T2 1 + 13.4iT - 71T^{2}
73 1+(−5.05−5.05i)T+73iT2 1 + (-5.05 - 5.05i)T + 73iT^{2}
79 1−8.85iT−79T2 1 - 8.85iT - 79T^{2}
83 1+(−4.78−4.78i)T+83iT2 1 + (-4.78 - 4.78i)T + 83iT^{2}
89 1+8.33T+89T2 1 + 8.33T + 89T^{2}
97 1+(−10.1+10.1i)T−97iT2 1 + (-10.1 + 10.1i)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

−11.10633810362802043859860649921, −10.66224784085880569939094641371, −9.439541861096698881695914750872, −8.396428030334200418574711202658, −7.85801552729013008169216947161, −6.90436998827225347246726111874, −5.78512490731491976090010770354, −4.67886228786382255856291890035, −3.40842976719496254791259290379, −0.936352854525418942323758779444, 1.27460084179640595879080310695, 3.12244489536911086679350650459, 3.99527879228947865417886421014, 5.26126941834362419345319144295, 7.08620829015100287196295018650, 7.71724160075939784583180983104, 8.903150608598766025290968668591, 9.366760577988439595400116824767, 10.76229608949856568242952485123, 11.42622275330600485062546946768

Graph of the ZZ-function along the critical line