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Properties

Label 2-630-105.2-c1-0-6
Degree 22
Conductor 630630
Sign 0.912−0.409i0.912 - 0.409i
Analytic cond. 5.030575.03057
Root an. cond. 2.242892.24289
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−2.22 + 0.227i)5-s + (1.18 − 2.36i)7-s + (−0.707 − 0.707i)8-s + (−0.795 − 2.08i)10-s + (1.10 − 0.636i)11-s + (2.15 − 2.15i)13-s + (2.59 + 0.537i)14-s + (0.500 − 0.866i)16-s + (5.80 + 1.55i)17-s + (6.20 + 3.58i)19-s + (1.81 − 1.30i)20-s + (0.900 + 0.900i)22-s + (−4.14 + 1.11i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.994 + 0.101i)5-s + (0.449 − 0.893i)7-s + (−0.249 − 0.249i)8-s + (−0.251 − 0.660i)10-s + (0.332 − 0.191i)11-s + (0.596 − 0.596i)13-s + (0.692 + 0.143i)14-s + (0.125 − 0.216i)16-s + (1.40 + 0.376i)17-s + (1.42 + 0.822i)19-s + (0.405 − 0.292i)20-s + (0.191 + 0.191i)22-s + (−0.864 + 0.231i)23-s + ⋯

Functional equation

Λ(s)=(630s/2 ΓC(s) L(s)=((0.912−0.409i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+1/2) L(s)=((0.912−0.409i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 630630    =    2⋅32⋅5⋅72 \cdot 3^{2} \cdot 5 \cdot 7
Sign: 0.912−0.409i0.912 - 0.409i
Analytic conductor: 5.030575.03057
Root analytic conductor: 2.242892.24289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ630(107,⋅)\chi_{630} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 630, ( :1/2), 0.912−0.409i)(2,\ 630,\ (\ :1/2),\ 0.912 - 0.409i)

Particular Values

L(1)L(1) ≈\approx 1.42187+0.304809i1.42187 + 0.304809i
L(12)L(\frac12) ≈\approx 1.42187+0.304809i1.42187 + 0.304809i
L(32)L(\frac{3}{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+(−0.258−0.965i)T 1 + (-0.258 - 0.965i)T
3 1 1
5 1+(2.22−0.227i)T 1 + (2.22 - 0.227i)T
7 1+(−1.18+2.36i)T 1 + (-1.18 + 2.36i)T
good11 1+(−1.10+0.636i)T+(5.5−9.52i)T2 1 + (-1.10 + 0.636i)T + (5.5 - 9.52i)T^{2}
13 1+(−2.15+2.15i)T−13iT2 1 + (-2.15 + 2.15i)T - 13iT^{2}
17 1+(−5.80−1.55i)T+(14.7+8.5i)T2 1 + (-5.80 - 1.55i)T + (14.7 + 8.5i)T^{2}
19 1+(−6.20−3.58i)T+(9.5+16.4i)T2 1 + (-6.20 - 3.58i)T + (9.5 + 16.4i)T^{2}
23 1+(4.14−1.11i)T+(19.9−11.5i)T2 1 + (4.14 - 1.11i)T + (19.9 - 11.5i)T^{2}
29 1−1.25T+29T2 1 - 1.25T + 29T^{2}
31 1+(2.35+4.08i)T+(−15.5+26.8i)T2 1 + (2.35 + 4.08i)T + (-15.5 + 26.8i)T^{2}
37 1+(1.99−0.535i)T+(32.0−18.5i)T2 1 + (1.99 - 0.535i)T + (32.0 - 18.5i)T^{2}
41 1+0.655iT−41T2 1 + 0.655iT - 41T^{2}
43 1+(−7.20+7.20i)T−43iT2 1 + (-7.20 + 7.20i)T - 43iT^{2}
47 1+(−2.80−10.4i)T+(−40.7+23.5i)T2 1 + (-2.80 - 10.4i)T + (-40.7 + 23.5i)T^{2}
53 1+(−3.55+13.2i)T+(−45.8−26.5i)T2 1 + (-3.55 + 13.2i)T + (-45.8 - 26.5i)T^{2}
59 1+(−0.688−1.19i)T+(−29.5+51.0i)T2 1 + (-0.688 - 1.19i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.57−4.46i)T+(−30.5−52.8i)T2 1 + (2.57 - 4.46i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.12+4.19i)T+(−58.0−33.5i)T2 1 + (-1.12 + 4.19i)T + (-58.0 - 33.5i)T^{2}
71 1−0.159iT−71T2 1 - 0.159iT - 71T^{2}
73 1+(−9.90−2.65i)T+(63.2+36.5i)T2 1 + (-9.90 - 2.65i)T + (63.2 + 36.5i)T^{2}
79 1+(9.23+5.33i)T+(39.5+68.4i)T2 1 + (9.23 + 5.33i)T + (39.5 + 68.4i)T^{2}
83 1+(6.09+6.09i)T+83iT2 1 + (6.09 + 6.09i)T + 83iT^{2}
89 1+(−3.79+6.57i)T+(−44.5−77.0i)T2 1 + (-3.79 + 6.57i)T + (-44.5 - 77.0i)T^{2}
97 1+(−10.9−10.9i)T+97iT2 1 + (-10.9 - 10.9i)T + 97iT^{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

−10.61974182351100371896312296014, −9.835138256073864876151901587046, −8.569078585936024924695661957907, −7.66605501592443047338459426174, −7.52328447882179769904009342969, −6.12583869411009859825347608232, −5.23370703769035478869620665979, −3.92442920928176887944129410703, −3.47660047911764005673170106334, −1.00144237195915450426852081820, 1.22696977177783757529947107870, 2.80989044325434875284034648782, 3.81240382978707089818980717818, 4.86539174386239332140151774495, 5.73771076013078551985044525742, 7.12046217146979754075663376557, 8.041191595434816911359700999633, 8.912611447926152015640648737116, 9.613108249540152748062979423348, 10.77880238331969175750697249410

Graph of the ZZ-function along the critical line