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Properties

Label 2-630-15.2-c3-0-23
Degree 22
Conductor 630630
Sign 0.0947+0.995i0.0947 + 0.995i
Analytic cond. 37.171237.1712
Root an. cond. 6.096816.09681
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  + (−1.41 + 1.41i)2-s − 4.00i·4-s + (1.74 + 11.0i)5-s + (−4.94 − 4.94i)7-s + (5.65 + 5.65i)8-s + (−18.0 − 13.1i)10-s + 2.28i·11-s + (25.7 − 25.7i)13-s + 14.0·14-s − 16.0·16-s + (−71.5 + 71.5i)17-s − 50.6i·19-s + (44.1 − 6.98i)20-s + (−3.23 − 3.23i)22-s + (−53.7 − 53.7i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.156 + 0.987i)5-s + (−0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + (−0.571 − 0.415i)10-s + 0.0627i·11-s + (0.549 − 0.549i)13-s + 0.267·14-s − 0.250·16-s + (−1.02 + 1.02i)17-s − 0.611i·19-s + (0.493 − 0.0780i)20-s + (−0.0313 − 0.0313i)22-s + (−0.487 − 0.487i)23-s + ⋯

Functional equation

Λ(s)=(630s/2 ΓC(s) L(s)=((0.0947+0.995i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0947 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+3/2) L(s)=((0.0947+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0947 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 630630    =    2⋅32⋅5⋅72 \cdot 3^{2} \cdot 5 \cdot 7
Sign: 0.0947+0.995i0.0947 + 0.995i
Analytic conductor: 37.171237.1712
Root analytic conductor: 6.096816.09681
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ630(197,⋅)\chi_{630} (197, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 630, ( :3/2), 0.0947+0.995i)(2,\ 630,\ (\ :3/2),\ 0.0947 + 0.995i)

Particular Values

L(2)L(2) ≈\approx 0.46435490600.4643549060
L(12)L(\frac12) ≈\approx 0.46435490600.4643549060
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+(1.41−1.41i)T 1 + (1.41 - 1.41i)T
3 1 1
5 1+(−1.74−11.0i)T 1 + (-1.74 - 11.0i)T
7 1+(4.94+4.94i)T 1 + (4.94 + 4.94i)T
good11 1−2.28iT−1.33e3T2 1 - 2.28iT - 1.33e3T^{2}
13 1+(−25.7+25.7i)T−2.19e3iT2 1 + (-25.7 + 25.7i)T - 2.19e3iT^{2}
17 1+(71.5−71.5i)T−4.91e3iT2 1 + (71.5 - 71.5i)T - 4.91e3iT^{2}
19 1+50.6iT−6.85e3T2 1 + 50.6iT - 6.85e3T^{2}
23 1+(53.7+53.7i)T+1.21e4iT2 1 + (53.7 + 53.7i)T + 1.21e4iT^{2}
29 1−186.T+2.43e4T2 1 - 186.T + 2.43e4T^{2}
31 1+211.T+2.97e4T2 1 + 211.T + 2.97e4T^{2}
37 1+(70.1+70.1i)T+5.06e4iT2 1 + (70.1 + 70.1i)T + 5.06e4iT^{2}
41 1−50.2iT−6.89e4T2 1 - 50.2iT - 6.89e4T^{2}
43 1+(102.−102.i)T−7.95e4iT2 1 + (102. - 102. i)T - 7.95e4iT^{2}
47 1+(−216.+216.i)T−1.03e5iT2 1 + (-216. + 216. i)T - 1.03e5iT^{2}
53 1+(91.3+91.3i)T+1.48e5iT2 1 + (91.3 + 91.3i)T + 1.48e5iT^{2}
59 1+234.T+2.05e5T2 1 + 234.T + 2.05e5T^{2}
61 1+22.8T+2.26e5T2 1 + 22.8T + 2.26e5T^{2}
67 1+(267.+267.i)T+3.00e5iT2 1 + (267. + 267. i)T + 3.00e5iT^{2}
71 1+408.iT−3.57e5T2 1 + 408. iT - 3.57e5T^{2}
73 1+(−15.6+15.6i)T−3.89e5iT2 1 + (-15.6 + 15.6i)T - 3.89e5iT^{2}
79 1+426.iT−4.93e5T2 1 + 426. iT - 4.93e5T^{2}
83 1+(659.+659.i)T+5.71e5iT2 1 + (659. + 659. i)T + 5.71e5iT^{2}
89 1−331.T+7.04e5T2 1 - 331.T + 7.04e5T^{2}
97 1+(875.+875.i)T+9.12e5iT2 1 + (875. + 875. i)T + 9.12e5iT^{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

−10.12172176439441190338755308060, −9.031968330032814626090029480245, −8.232476321651897297252485468079, −7.22993630211233663035638046070, −6.50271297928474964528619462358, −5.80627231864993766567571035175, −4.38163427750618067184688690136, −3.18077789888685880537899688313, −1.88779999494582324714239831690, −0.16526238789031400076098982499, 1.21120758590675323706953763806, 2.34332134572628696854141307365, 3.74434119572147679118435840124, 4.74375301065508561799683580929, 5.85092062035341653980544618605, 6.93570224175945804132488563914, 8.056498816326432643178856872308, 8.899692771847225042811626970595, 9.356801291102008581271066950192, 10.29148584329868389919913922681

Graph of the ZZ-function along the critical line