↑

Properties

Label 2-630-35.17-c1-0-8
Degree 22
Conductor 630630
Sign −0.0172−0.999i-0.0172 - 0.999i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.774 + 2.09i)5-s + (2.64 + 0.126i)7-s + (0.707 − 0.707i)8-s + (−1.82 − 1.29i)10-s + (2.81 − 4.87i)11-s + (1.42 + 1.42i)13-s + (−0.806 + 2.51i)14-s + (0.500 + 0.866i)16-s + (1.37 + 5.12i)17-s + (1.94 + 3.37i)19-s + (1.71 − 1.42i)20-s + (3.97 + 3.97i)22-s + (−1.08 − 0.290i)23-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (−0.346 + 0.938i)5-s + (0.998 + 0.0477i)7-s + (0.249 − 0.249i)8-s + (−0.577 − 0.408i)10-s + (0.848 − 1.46i)11-s + (0.396 + 0.396i)13-s + (−0.215 + 0.673i)14-s + (0.125 + 0.216i)16-s + (0.333 + 1.24i)17-s + (0.446 + 0.773i)19-s + (0.384 − 0.319i)20-s + (0.848 + 0.848i)22-s + (−0.226 − 0.0606i)23-s + ⋯

Functional equation

Λ(s)=(630s/2 ΓC(s) L(s)=((−0.0172−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0172 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+1/2) L(s)=((−0.0172−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0172 - 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: −0.0172−0.999i-0.0172 - 0.999i
Analytic conductor: 5.030575.03057
Root analytic conductor: 2.242892.24289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ630(577,⋅)\chi_{630} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 630, ( :1/2), −0.0172−0.999i)(2,\ 630,\ (\ :1/2),\ -0.0172 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.985064+1.00216i0.985064 + 1.00216i
L(12)L(\frac12) ≈\approx 0.985064+1.00216i0.985064 + 1.00216i
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+(0.774−2.09i)T 1 + (0.774 - 2.09i)T
7 1+(−2.64−0.126i)T 1 + (-2.64 - 0.126i)T
good11 1+(−2.81+4.87i)T+(−5.5−9.52i)T2 1 + (-2.81 + 4.87i)T + (-5.5 - 9.52i)T^{2}
13 1+(−1.42−1.42i)T+13iT2 1 + (-1.42 - 1.42i)T + 13iT^{2}
17 1+(−1.37−5.12i)T+(−14.7+8.5i)T2 1 + (-1.37 - 5.12i)T + (-14.7 + 8.5i)T^{2}
19 1+(−1.94−3.37i)T+(−9.5+16.4i)T2 1 + (-1.94 - 3.37i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.08+0.290i)T+(19.9+11.5i)T2 1 + (1.08 + 0.290i)T + (19.9 + 11.5i)T^{2}
29 1−3.15iT−29T2 1 - 3.15iT - 29T^{2}
31 1+(3.33+1.92i)T+(15.5+26.8i)T2 1 + (3.33 + 1.92i)T + (15.5 + 26.8i)T^{2}
37 1+(1.30−4.86i)T+(−32.0−18.5i)T2 1 + (1.30 - 4.86i)T + (-32.0 - 18.5i)T^{2}
41 1−7.21iT−41T2 1 - 7.21iT - 41T^{2}
43 1+(−1.85+1.85i)T−43iT2 1 + (-1.85 + 1.85i)T - 43iT^{2}
47 1+(−5.69−1.52i)T+(40.7+23.5i)T2 1 + (-5.69 - 1.52i)T + (40.7 + 23.5i)T^{2}
53 1+(−0.357−1.33i)T+(−45.8+26.5i)T2 1 + (-0.357 - 1.33i)T + (-45.8 + 26.5i)T^{2}
59 1+(−2.73+4.74i)T+(−29.5−51.0i)T2 1 + (-2.73 + 4.74i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.99−2.30i)T+(30.5−52.8i)T2 1 + (3.99 - 2.30i)T + (30.5 - 52.8i)T^{2}
67 1+(0.816−0.218i)T+(58.0−33.5i)T2 1 + (0.816 - 0.218i)T + (58.0 - 33.5i)T^{2}
71 1+4.77T+71T2 1 + 4.77T + 71T^{2}
73 1+(−5.42+1.45i)T+(63.2−36.5i)T2 1 + (-5.42 + 1.45i)T + (63.2 - 36.5i)T^{2}
79 1+(5.41−3.12i)T+(39.5−68.4i)T2 1 + (5.41 - 3.12i)T + (39.5 - 68.4i)T^{2}
83 1+(−5.67−5.67i)T+83iT2 1 + (-5.67 - 5.67i)T + 83iT^{2}
89 1+(5.96+10.3i)T+(−44.5+77.0i)T2 1 + (5.96 + 10.3i)T + (-44.5 + 77.0i)T^{2}
97 1+(−6.63+6.63i)T−97iT2 1 + (-6.63 + 6.63i)T - 97iT^{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

−10.92659252363954936731699653294, −9.967428549052866758273818056405, −8.717991328484331507905214352921, −8.217677378557810328303695363511, −7.34135389251959524382764142856, −6.26329108023602811459792059204, −5.71895348150591538054855325057, −4.20859617561997475798694958496, −3.37326789717661710567951912375, −1.47045975906220467538209455184, 0.981609851928948372742980243870, 2.16919217681946424645019821924, 3.84484106946892002421912132107, 4.68134699030956098353025335725, 5.41245640177877183644882085618, 7.18656524471831663274254430248, 7.77627711295113057650529643833, 8.941889819723380020863360434103, 9.342208485940641554802909650669, 10.40493713661763569198280469006

Graph of the ZZ-function along the critical line