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Properties

Label 2-630-15.2-c3-0-26
Degree 22
Conductor 630630
Sign −0.00354+0.999i-0.00354 + 0.999i
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)=((−0.00354+0.999i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00354 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+3/2) L(s)=((−0.00354+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00354 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 630630    =    2⋅32⋅5⋅72 \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.00354+0.999i-0.00354 + 0.999i
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), −0.00354+0.999i)(2,\ 630,\ (\ :3/2),\ -0.00354 + 0.999i)

Particular Values

L(2)L(2) ≈\approx 3.2407022233.240702223
L(12)L(\frac12) ≈\approx 3.2407022233.240702223
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

−10.01258400737511696408845407480, −9.278604681347568811879123872695, −8.462726809929083686096531878875, −7.14444151055415215241553269349, −6.14308759414238664718317348138, −5.22297100660050189721570401457, −4.54865883486199647168656748027, −3.04807293144207572191752442955, −2.07305901798702986796856418952, −0.842646020015587793720714513357, 1.38972346293161601953375863994, 2.75795646430239163271116289438, 3.88081058527699257562177955452, 5.06077784602578624468133320279, 5.96294054608124450835798817446, 6.61612430667934373589744249303, 7.70529715780556879504814246076, 8.496516652148740330231168792519, 9.615842781526480678562465340783, 10.38699202802616365495355130726

Graph of the ZZ-function along the critical line