↑

Properties

Label 2-630-105.2-c1-0-7
Degree 22
Conductor 630630
Sign 0.840+0.542i0.840 + 0.542i
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 + (1.48 + 1.67i)5-s + (1.30 − 2.30i)7-s + (0.707 + 0.707i)8-s + (1.23 − 1.86i)10-s + (3.21 − 1.85i)11-s + (−4.62 + 4.62i)13-s + (−2.56 − 0.662i)14-s + (0.500 − 0.866i)16-s + (5.06 + 1.35i)17-s + (−0.866 − 0.5i)19-s + (−2.12 − 0.707i)20-s + (−2.62 − 2.62i)22-s + (4.10 − 1.09i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.663 + 0.748i)5-s + (0.492 − 0.870i)7-s + (0.249 + 0.249i)8-s + (0.389 − 0.590i)10-s + (0.968 − 0.559i)11-s + (−1.28 + 1.28i)13-s + (−0.684 − 0.177i)14-s + (0.125 − 0.216i)16-s + (1.22 + 0.329i)17-s + (−0.198 − 0.114i)19-s + (−0.474 − 0.158i)20-s + (−0.559 − 0.559i)22-s + (0.854 − 0.229i)23-s + ⋯

Functional equation

Λ(s)=(630s/2 ΓC(s) L(s)=((0.840+0.542i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+1/2) L(s)=((0.840+0.542i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 630630    =    2⋅32⋅5⋅72 \cdot 3^{2} \cdot 5 \cdot 7
Sign: 0.840+0.542i0.840 + 0.542i
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), 0.840+0.542i)(2,\ 630,\ (\ :1/2),\ 0.840 + 0.542i)

Particular Values

L(1)L(1) ≈\approx 1.51847−0.447621i1.51847 - 0.447621i
L(12)L(\frac12) ≈\approx 1.51847−0.447621i1.51847 - 0.447621i
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+(−1.48−1.67i)T 1 + (-1.48 - 1.67i)T
7 1+(−1.30+2.30i)T 1 + (-1.30 + 2.30i)T
good11 1+(−3.21+1.85i)T+(5.5−9.52i)T2 1 + (-3.21 + 1.85i)T + (5.5 - 9.52i)T^{2}
13 1+(4.62−4.62i)T−13iT2 1 + (4.62 - 4.62i)T - 13iT^{2}
17 1+(−5.06−1.35i)T+(14.7+8.5i)T2 1 + (-5.06 - 1.35i)T + (14.7 + 8.5i)T^{2}
19 1+(0.866+0.5i)T+(9.5+16.4i)T2 1 + (0.866 + 0.5i)T + (9.5 + 16.4i)T^{2}
23 1+(−4.10+1.09i)T+(19.9−11.5i)T2 1 + (-4.10 + 1.09i)T + (19.9 - 11.5i)T^{2}
29 1−2.82T+29T2 1 - 2.82T + 29T^{2}
31 1+(2+3.46i)T+(−15.5+26.8i)T2 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2}
37 1+(−10.4+2.79i)T+(32.0−18.5i)T2 1 + (-10.4 + 2.79i)T + (32.0 - 18.5i)T^{2}
41 1+0.880iT−41T2 1 + 0.880iT - 41T^{2}
43 1+(−3+3i)T−43iT2 1 + (-3 + 3i)T - 43iT^{2}
47 1+(−2.32−8.69i)T+(−40.7+23.5i)T2 1 + (-2.32 - 8.69i)T + (-40.7 + 23.5i)T^{2}
53 1+(0.581−2.16i)T+(−45.8−26.5i)T2 1 + (0.581 - 2.16i)T + (-45.8 - 26.5i)T^{2}
59 1+(5.83+10.0i)T+(−29.5+51.0i)T2 1 + (5.83 + 10.0i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.62+2.81i)T+(−30.5−52.8i)T2 1 + (-1.62 + 2.81i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.56−9.56i)T+(−58.0−33.5i)T2 1 + (2.56 - 9.56i)T + (-58.0 - 33.5i)T^{2}
71 1+10.2iT−71T2 1 + 10.2iT - 71T^{2}
73 1+(13.9+3.74i)T+(63.2+36.5i)T2 1 + (13.9 + 3.74i)T + (63.2 + 36.5i)T^{2}
79 1+(−2.38−1.37i)T+(39.5+68.4i)T2 1 + (-2.38 - 1.37i)T + (39.5 + 68.4i)T^{2}
83 1+(−6.53−6.53i)T+83iT2 1 + (-6.53 - 6.53i)T + 83iT^{2}
89 1+(4.94−8.57i)T+(−44.5−77.0i)T2 1 + (4.94 - 8.57i)T + (-44.5 - 77.0i)T^{2}
97 1+(9.24+9.24i)T+97iT2 1 + (9.24 + 9.24i)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

−10.59938090921911019783539575131, −9.656723269724921851003245979681, −9.188081186110901066846652014196, −7.81786565972850134982002646247, −7.05163649252767264676659925687, −6.07816850433543122173115654384, −4.72880292653376587705904861964, −3.77904875606644749474743056300, −2.54392219225770456374796043151, −1.28861114684287750746198181325, 1.23907272182159115839751284693, 2.76372674786876487565830839044, 4.58221377063638498597194088901, 5.29848184062854407323399278779, 5.98762595824786683554605677225, 7.24048888169448116789166990883, 8.058213035117390180818985566135, 8.949031978069033470377644629528, 9.633713431805765402068448270464, 10.29128915455658227900741816171

Graph of the ZZ-function along the critical line