Properties

Label 2-360-15.8-c3-0-3
Degree 22
Conductor 360360
Sign 0.4800.877i-0.480 - 0.877i
Analytic cond. 21.240621.2406
Root an. cond. 4.608764.60876
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.08 + 8.64i)5-s + (3.01 − 3.01i)7-s − 64.6i·11-s + (46.4 + 46.4i)13-s + (51.1 + 51.1i)17-s + 75.4i·19-s + (−131. + 131. i)23-s + (−24.6 − 122. i)25-s − 279.·29-s − 7.69·31-s + (4.72 + 47.5i)35-s + (−152. + 152. i)37-s + 295. i·41-s + (200. + 200. i)43-s + (−292. − 292. i)47-s + ⋯
L(s)  = 1  + (−0.633 + 0.773i)5-s + (0.163 − 0.163i)7-s − 1.77i·11-s + (0.992 + 0.992i)13-s + (0.730 + 0.730i)17-s + 0.911i·19-s + (−1.19 + 1.19i)23-s + (−0.196 − 0.980i)25-s − 1.78·29-s − 0.0445·31-s + (0.0228 + 0.229i)35-s + (−0.678 + 0.678i)37-s + 1.12i·41-s + (0.710 + 0.710i)43-s + (−0.907 − 0.907i)47-s + ⋯

Functional equation

Λ(s)=(360s/2ΓC(s)L(s)=((0.4800.877i)Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(360s/2ΓC(s+3/2)L(s)=((0.4800.877i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    233252^{3} \cdot 3^{2} \cdot 5
Sign: 0.4800.877i-0.480 - 0.877i
Analytic conductor: 21.240621.2406
Root analytic conductor: 4.608764.60876
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ360(233,)\chi_{360} (233, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :3/2), 0.4800.877i)(2,\ 360,\ (\ :3/2),\ -0.480 - 0.877i)

Particular Values

L(2)L(2) \approx 1.0740877051.074087705
L(12)L(\frac12) \approx 1.0740877051.074087705
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
5 1+(7.088.64i)T 1 + (7.08 - 8.64i)T
good7 1+(3.01+3.01i)T343iT2 1 + (-3.01 + 3.01i)T - 343iT^{2}
11 1+64.6iT1.33e3T2 1 + 64.6iT - 1.33e3T^{2}
13 1+(46.446.4i)T+2.19e3iT2 1 + (-46.4 - 46.4i)T + 2.19e3iT^{2}
17 1+(51.151.1i)T+4.91e3iT2 1 + (-51.1 - 51.1i)T + 4.91e3iT^{2}
19 175.4iT6.85e3T2 1 - 75.4iT - 6.85e3T^{2}
23 1+(131.131.i)T1.21e4iT2 1 + (131. - 131. i)T - 1.21e4iT^{2}
29 1+279.T+2.43e4T2 1 + 279.T + 2.43e4T^{2}
31 1+7.69T+2.97e4T2 1 + 7.69T + 2.97e4T^{2}
37 1+(152.152.i)T5.06e4iT2 1 + (152. - 152. i)T - 5.06e4iT^{2}
41 1295.iT6.89e4T2 1 - 295. iT - 6.89e4T^{2}
43 1+(200.200.i)T+7.95e4iT2 1 + (-200. - 200. i)T + 7.95e4iT^{2}
47 1+(292.+292.i)T+1.03e5iT2 1 + (292. + 292. i)T + 1.03e5iT^{2}
53 1+(212.212.i)T1.48e5iT2 1 + (212. - 212. i)T - 1.48e5iT^{2}
59 1+55.6T+2.05e5T2 1 + 55.6T + 2.05e5T^{2}
61 119.2T+2.26e5T2 1 - 19.2T + 2.26e5T^{2}
67 1+(437.+437.i)T3.00e5iT2 1 + (-437. + 437. i)T - 3.00e5iT^{2}
71 1582.iT3.57e5T2 1 - 582. iT - 3.57e5T^{2}
73 1+(172.+172.i)T+3.89e5iT2 1 + (172. + 172. i)T + 3.89e5iT^{2}
79 1767.iT4.93e5T2 1 - 767. iT - 4.93e5T^{2}
83 1+(369.+369.i)T5.71e5iT2 1 + (-369. + 369. i)T - 5.71e5iT^{2}
89 1394.T+7.04e5T2 1 - 394.T + 7.04e5T^{2}
97 1+(291.291.i)T9.12e5iT2 1 + (291. - 291. i)T - 9.12e5iT^{2}
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   L(s)=p j=12(1αj,pps)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

−11.30280207091846633208830039004, −10.65134215951136505343434211013, −9.498144221336004029887619425035, −8.280671328786790633535521250052, −7.77145432253395873453379381751, −6.37895371652999454627907021396, −5.75594916142004425471827716725, −3.87805787783619885301149181654, −3.41099356271653098110444985021, −1.49656815164524061320103505552, 0.38436512624979855600143019174, 1.98529662302810052589802718871, 3.66928289639470738223035424319, 4.72101251344881296527629648789, 5.62542505747591778848919516700, 7.12714974057478889152223865027, 7.85896077318672755315086188889, 8.837728898586267877477508341284, 9.721664733791871697666499226180, 10.73806671334384533963576421934

Graph of the ZZ-function along the critical line