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Properties

Label 2-630-15.2-c3-0-17
Degree 22
Conductor 630630
Sign 0.943−0.329i0.943 - 0.329i
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 + (−10.3 + 4.33i)5-s + (4.94 + 4.94i)7-s + (5.65 + 5.65i)8-s + (8.44 − 20.7i)10-s − 8.72i·11-s + (3.39 − 3.39i)13-s − 14.0·14-s − 16.0·16-s + (−35.6 + 35.6i)17-s − 87.4i·19-s + (17.3 + 41.2i)20-s + (12.3 + 12.3i)22-s + (−8.19 − 8.19i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.921 + 0.387i)5-s + (0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + (0.267 − 0.654i)10-s − 0.239i·11-s + (0.0724 − 0.0724i)13-s − 0.267·14-s − 0.250·16-s + (−0.508 + 0.508i)17-s − 1.05i·19-s + (0.193 + 0.460i)20-s + (0.119 + 0.119i)22-s + (−0.0743 − 0.0743i)23-s + ⋯

Functional equation

Λ(s)=(630s/2 ΓC(s) L(s)=((0.943−0.329i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.329i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+3/2) L(s)=((0.943−0.329i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.943 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 630630    =    2⋅32⋅5⋅72 \cdot 3^{2} \cdot 5 \cdot 7
Sign: 0.943−0.329i0.943 - 0.329i
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), 0.943−0.329i)(2,\ 630,\ (\ :3/2),\ 0.943 - 0.329i)

Particular Values

L(2)L(2) ≈\approx 1.0234912401.023491240
L(12)L(\frac12) ≈\approx 1.0234912401.023491240
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+(10.3−4.33i)T 1 + (10.3 - 4.33i)T
7 1+(−4.94−4.94i)T 1 + (-4.94 - 4.94i)T
good11 1+8.72iT−1.33e3T2 1 + 8.72iT - 1.33e3T^{2}
13 1+(−3.39+3.39i)T−2.19e3iT2 1 + (-3.39 + 3.39i)T - 2.19e3iT^{2}
17 1+(35.6−35.6i)T−4.91e3iT2 1 + (35.6 - 35.6i)T - 4.91e3iT^{2}
19 1+87.4iT−6.85e3T2 1 + 87.4iT - 6.85e3T^{2}
23 1+(8.19+8.19i)T+1.21e4iT2 1 + (8.19 + 8.19i)T + 1.21e4iT^{2}
29 1+199.T+2.43e4T2 1 + 199.T + 2.43e4T^{2}
31 1+21.6T+2.97e4T2 1 + 21.6T + 2.97e4T^{2}
37 1+(−6.63−6.63i)T+5.06e4iT2 1 + (-6.63 - 6.63i)T + 5.06e4iT^{2}
41 1−95.4iT−6.89e4T2 1 - 95.4iT - 6.89e4T^{2}
43 1+(−144.+144.i)T−7.95e4iT2 1 + (-144. + 144. i)T - 7.95e4iT^{2}
47 1+(−30.7+30.7i)T−1.03e5iT2 1 + (-30.7 + 30.7i)T - 1.03e5iT^{2}
53 1+(−221.−221.i)T+1.48e5iT2 1 + (-221. - 221. i)T + 1.48e5iT^{2}
59 1−531.T+2.05e5T2 1 - 531.T + 2.05e5T^{2}
61 1−578.T+2.26e5T2 1 - 578.T + 2.26e5T^{2}
67 1+(−222.−222.i)T+3.00e5iT2 1 + (-222. - 222. i)T + 3.00e5iT^{2}
71 1+134.iT−3.57e5T2 1 + 134. iT - 3.57e5T^{2}
73 1+(−111.+111.i)T−3.89e5iT2 1 + (-111. + 111. i)T - 3.89e5iT^{2}
79 1+1.23e3iT−4.93e5T2 1 + 1.23e3iT - 4.93e5T^{2}
83 1+(−336.−336.i)T+5.71e5iT2 1 + (-336. - 336. i)T + 5.71e5iT^{2}
89 1−509.T+7.04e5T2 1 - 509.T + 7.04e5T^{2}
97 1+(−408.−408.i)T+9.12e5iT2 1 + (-408. - 408. 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

−10.28113301201416396577629763657, −9.099830805426542172251693750295, −8.491760111781366537566174424597, −7.58632926104814931003367874539, −6.88412095771203095684512733119, −5.86648161581925251700144462298, −4.74232824419703786093163969366, −3.65428122616987019791839751397, −2.26615890543214910229792300003, −0.56672771322763028923409433159, 0.71420673217140635305481891297, 2.03884604510448222768970369539, 3.53480297453383336763306805469, 4.28623338792144656192041694261, 5.43182488335245400985163980621, 6.91593368509429567592984612920, 7.68268795014470296864304466564, 8.418929445375514823264728874810, 9.274012658655828097823452338554, 10.15726688517056861065334902044

Graph of the ZZ-function along the critical line