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Properties

Label 2-360-120.59-c1-0-14
Degree 22
Conductor 360360
Sign 0.603+0.797i0.603 + 0.797i
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  + (−1.37 − 0.331i)2-s + (1.78 + 0.910i)4-s + (1.64 − 1.51i)5-s + 0.936·7-s + (−2.14 − 1.84i)8-s + (−2.76 + 1.53i)10-s − 2.20i·11-s + 3.33·13-s + (−1.28 − 0.310i)14-s + (2.34 + 3.24i)16-s − 1.54·17-s − 3.12·19-s + (4.31 − 1.18i)20-s + (−0.731 + 3.03i)22-s + 3.39i·23-s + ⋯
L(s)  = 1  + (−0.972 − 0.234i)2-s + (0.890 + 0.455i)4-s + (0.737 − 0.675i)5-s + 0.353·7-s + (−0.759 − 0.650i)8-s + (−0.875 + 0.483i)10-s − 0.665i·11-s + 0.924·13-s + (−0.344 − 0.0828i)14-s + (0.585 + 0.810i)16-s − 0.374·17-s − 0.716·19-s + (0.964 − 0.265i)20-s + (−0.155 + 0.647i)22-s + 0.707i·23-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((0.603+0.797i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+1/2) L(s)=((0.603+0.797i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: 0.603+0.797i0.603 + 0.797i
Analytic conductor: 2.874612.87461
Root analytic conductor: 1.695461.69546
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ360(179,⋅)\chi_{360} (179, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :1/2), 0.603+0.797i)(2,\ 360,\ (\ :1/2),\ 0.603 + 0.797i)

Particular Values

L(1)L(1) ≈\approx 0.930989−0.462834i0.930989 - 0.462834i
L(12)L(\frac12) ≈\approx 0.930989−0.462834i0.930989 - 0.462834i
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+(1.37+0.331i)T 1 + (1.37 + 0.331i)T
3 1 1
5 1+(−1.64+1.51i)T 1 + (-1.64 + 1.51i)T
good7 1−0.936T+7T2 1 - 0.936T + 7T^{2}
11 1+2.20iT−11T2 1 + 2.20iT - 11T^{2}
13 1−3.33T+13T2 1 - 3.33T + 13T^{2}
17 1+1.54T+17T2 1 + 1.54T + 17T^{2}
19 1+3.12T+19T2 1 + 3.12T + 19T^{2}
23 1−3.39iT−23T2 1 - 3.39iT - 23T^{2}
29 1−8.44T+29T2 1 - 8.44T + 29T^{2}
31 1+8.30iT−31T2 1 + 8.30iT - 31T^{2}
37 1−7.60T+37T2 1 - 7.60T + 37T^{2}
41 1+5.83iT−41T2 1 + 5.83iT - 41T^{2}
43 1+7.77iT−43T2 1 + 7.77iT - 43T^{2}
47 1−10.7iT−47T2 1 - 10.7iT - 47T^{2}
53 1+5.08iT−53T2 1 + 5.08iT - 53T^{2}
59 1−10.6iT−59T2 1 - 10.6iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1−12.1iT−67T2 1 - 12.1iT - 67T^{2}
71 1+11.7T+71T2 1 + 11.7T + 71T^{2}
73 1−5.59iT−73T2 1 - 5.59iT - 73T^{2}
79 1+1.02iT−79T2 1 + 1.02iT - 79T^{2}
83 1−14.0T+83T2 1 - 14.0T + 83T^{2}
89 1−13.0iT−89T2 1 - 13.0iT - 89T^{2}
97 1−2.18iT−97T2 1 - 2.18iT - 97T^{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

−11.13517502896787436159661703667, −10.35224192307352974266320962456, −9.344844670648950130854113286643, −8.632195492127046384920777036026, −7.925360457674972738311337989531, −6.49581414082637806121756328410, −5.72379553783218024242215926163, −4.12552380598186158490468204218, −2.49001092251217997144707563880, −1.10666746908855797198803650685, 1.62646792372811048951692543950, 2.88542075493203085944204256910, 4.81288647588673909455643605621, 6.27417417565894895621362699591, 6.68535945384832595423293101179, 7.965933928964309246935863025603, 8.791794066790340558251374376177, 9.785213533647176176815376471047, 10.56655776140896249823488319813, 11.15399833685619417019131939118

Graph of the ZZ-function along the critical line