↑

Properties

Label 2-360-15.8-c3-0-2
Degree 22
Conductor 360360
Sign −0.995+0.0922i-0.995 + 0.0922i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (5.12 + 9.93i)5-s + (−14.2 + 14.2i)7-s + 38.4i·11-s + (−11.0 − 11.0i)13-s + (−95.0 − 95.0i)17-s + 17.3i·19-s + (115. − 115. i)23-s + (−72.4 + 101. i)25-s − 193.·29-s − 107.·31-s + (−214. − 68.5i)35-s + (43.7 − 43.7i)37-s − 225. i·41-s + (306. + 306. i)43-s + (−220. − 220. i)47-s + ⋯
L(s)  = 1  + (0.458 + 0.888i)5-s + (−0.769 + 0.769i)7-s + 1.05i·11-s + (−0.235 − 0.235i)13-s + (−1.35 − 1.35i)17-s + 0.209i·19-s + (1.04 − 1.04i)23-s + (−0.579 + 0.814i)25-s − 1.24·29-s − 0.620·31-s + (−1.03 − 0.331i)35-s + (0.194 − 0.194i)37-s − 0.858i·41-s + (1.08 + 1.08i)43-s + (−0.684 − 0.684i)47-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((−0.995+0.0922i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0922i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+3/2) L(s)=((−0.995+0.0922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: −0.995+0.0922i-0.995 + 0.0922i
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), −0.995+0.0922i)(2,\ 360,\ (\ :3/2),\ -0.995 + 0.0922i)

Particular Values

L(2)L(2) ≈\approx 0.55696392970.5569639297
L(12)L(\frac12) ≈\approx 0.55696392970.5569639297
L(52)L(\frac{5}{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
3 1 1
5 1+(−5.12−9.93i)T 1 + (-5.12 - 9.93i)T
good7 1+(14.2−14.2i)T−343iT2 1 + (14.2 - 14.2i)T - 343iT^{2}
11 1−38.4iT−1.33e3T2 1 - 38.4iT - 1.33e3T^{2}
13 1+(11.0+11.0i)T+2.19e3iT2 1 + (11.0 + 11.0i)T + 2.19e3iT^{2}
17 1+(95.0+95.0i)T+4.91e3iT2 1 + (95.0 + 95.0i)T + 4.91e3iT^{2}
19 1−17.3iT−6.85e3T2 1 - 17.3iT - 6.85e3T^{2}
23 1+(−115.+115.i)T−1.21e4iT2 1 + (-115. + 115. i)T - 1.21e4iT^{2}
29 1+193.T+2.43e4T2 1 + 193.T + 2.43e4T^{2}
31 1+107.T+2.97e4T2 1 + 107.T + 2.97e4T^{2}
37 1+(−43.7+43.7i)T−5.06e4iT2 1 + (-43.7 + 43.7i)T - 5.06e4iT^{2}
41 1+225.iT−6.89e4T2 1 + 225. iT - 6.89e4T^{2}
43 1+(−306.−306.i)T+7.95e4iT2 1 + (-306. - 306. i)T + 7.95e4iT^{2}
47 1+(220.+220.i)T+1.03e5iT2 1 + (220. + 220. i)T + 1.03e5iT^{2}
53 1+(257.−257.i)T−1.48e5iT2 1 + (257. - 257. i)T - 1.48e5iT^{2}
59 1+124.T+2.05e5T2 1 + 124.T + 2.05e5T^{2}
61 1+862.T+2.26e5T2 1 + 862.T + 2.26e5T^{2}
67 1+(557.−557.i)T−3.00e5iT2 1 + (557. - 557. i)T - 3.00e5iT^{2}
71 1+246.iT−3.57e5T2 1 + 246. iT - 3.57e5T^{2}
73 1+(−77.9−77.9i)T+3.89e5iT2 1 + (-77.9 - 77.9i)T + 3.89e5iT^{2}
79 1−1.27e3iT−4.93e5T2 1 - 1.27e3iT - 4.93e5T^{2}
83 1+(−502.+502.i)T−5.71e5iT2 1 + (-502. + 502. i)T - 5.71e5iT^{2}
89 1+958.T+7.04e5T2 1 + 958.T + 7.04e5T^{2}
97 1+(−661.+661.i)T−9.12e5iT2 1 + (-661. + 661. i)T - 9.12e5iT^{2}
show more
show less
   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

−11.36841332265430455745714786361, −10.58537362060527284267393742551, −9.471791855581143032636792142587, −9.119038831729163678609171646286, −7.42756048503964122563293665950, −6.78689054658424486304065253827, −5.79530561683643125412480878396, −4.59582528572000506295546261829, −2.98869531204609194643081015006, −2.19589966553866214712575791394, 0.18124043086336825833957873445, 1.64256060782455956033257805963, 3.38090280694812707292876923513, 4.45772242118516664716730517812, 5.72135574858483758530110842494, 6.56087766419437513873147760663, 7.75093702172628306399751906742, 8.919365802507726300846623482270, 9.408981404976625362307316664444, 10.62011599490361583813017472182

Graph of the ZZ-function along the critical line