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Properties

Label 2-684-684.119-c1-0-89
Degree 22
Conductor 684684
Sign −0.997−0.0666i-0.997 - 0.0666i
Analytic cond. 5.461765.46176
Root an. cond. 2.337042.33704
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  + (−1.36 − 0.360i)2-s + (0.356 − 1.69i)3-s + (1.74 + 0.985i)4-s + (−1.53 − 0.270i)5-s + (−1.09 + 2.18i)6-s − 0.923i·7-s + (−2.02 − 1.97i)8-s + (−2.74 − 1.20i)9-s + (2.00 + 0.923i)10-s + (1.38 + 2.40i)11-s + (2.29 − 2.59i)12-s + (5.07 − 4.26i)13-s + (−0.332 + 1.26i)14-s + (−1.00 + 2.50i)15-s + (2.05 + 3.43i)16-s + (−0.788 − 2.16i)17-s + ⋯
L(s)  = 1  + (−0.966 − 0.254i)2-s + (0.205 − 0.978i)3-s + (0.870 + 0.492i)4-s + (−0.686 − 0.121i)5-s + (−0.448 + 0.893i)6-s − 0.349i·7-s + (−0.715 − 0.698i)8-s + (−0.915 − 0.402i)9-s + (0.632 + 0.291i)10-s + (0.418 + 0.725i)11-s + (0.661 − 0.750i)12-s + (1.40 − 1.18i)13-s + (−0.0889 + 0.337i)14-s + (−0.259 + 0.646i)15-s + (0.514 + 0.857i)16-s + (−0.191 − 0.525i)17-s + ⋯

Functional equation

Λ(s)=(684s/2 ΓC(s) L(s)=((−0.997−0.0666i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0666i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(684s/2 ΓC(s+1/2) L(s)=((−0.997−0.0666i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0666i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 684684    =    22⋅32⋅192^{2} \cdot 3^{2} \cdot 19
Sign: −0.997−0.0666i-0.997 - 0.0666i
Analytic conductor: 5.461765.46176
Root analytic conductor: 2.337042.33704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ684(119,⋅)\chi_{684} (119, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 684, ( :1/2), −0.997−0.0666i)(2,\ 684,\ (\ :1/2),\ -0.997 - 0.0666i)

Particular Values

L(1)L(1) ≈\approx 0.0184745+0.553715i0.0184745 + 0.553715i
L(12)L(\frac12) ≈\approx 0.0184745+0.553715i0.0184745 + 0.553715i
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+(1.36+0.360i)T 1 + (1.36 + 0.360i)T
3 1+(−0.356+1.69i)T 1 + (-0.356 + 1.69i)T
19 1+(3.66+2.36i)T 1 + (3.66 + 2.36i)T
good5 1+(1.53+0.270i)T+(4.69+1.71i)T2 1 + (1.53 + 0.270i)T + (4.69 + 1.71i)T^{2}
7 1+0.923iT−7T2 1 + 0.923iT - 7T^{2}
11 1+(−1.38−2.40i)T+(−5.5+9.52i)T2 1 + (-1.38 - 2.40i)T + (-5.5 + 9.52i)T^{2}
13 1+(−5.07+4.26i)T+(2.25−12.8i)T2 1 + (-5.07 + 4.26i)T + (2.25 - 12.8i)T^{2}
17 1+(0.788+2.16i)T+(−13.0+10.9i)T2 1 + (0.788 + 2.16i)T + (-13.0 + 10.9i)T^{2}
23 1+(8.65+3.15i)T+(17.6+14.7i)T2 1 + (8.65 + 3.15i)T + (17.6 + 14.7i)T^{2}
29 1+(−3.10−3.69i)T+(−5.03+28.5i)T2 1 + (-3.10 - 3.69i)T + (-5.03 + 28.5i)T^{2}
31 1+(−3.56−2.06i)T+(15.5+26.8i)T2 1 + (-3.56 - 2.06i)T + (15.5 + 26.8i)T^{2}
37 1+5.89T+37T2 1 + 5.89T + 37T^{2}
41 1+(−0.179−0.494i)T+(−31.4+26.3i)T2 1 + (-0.179 - 0.494i)T + (-31.4 + 26.3i)T^{2}
43 1+(2.04+5.62i)T+(−32.9+27.6i)T2 1 + (2.04 + 5.62i)T + (-32.9 + 27.6i)T^{2}
47 1+(2.45−2.05i)T+(8.16−46.2i)T2 1 + (2.45 - 2.05i)T + (8.16 - 46.2i)T^{2}
53 1+(3.83+4.57i)T+(−9.20+52.1i)T2 1 + (3.83 + 4.57i)T + (-9.20 + 52.1i)T^{2}
59 1+(2.01+1.69i)T+(10.2+58.1i)T2 1 + (2.01 + 1.69i)T + (10.2 + 58.1i)T^{2}
61 1+(−2.41−13.7i)T+(−57.3+20.8i)T2 1 + (-2.41 - 13.7i)T + (-57.3 + 20.8i)T^{2}
67 1+(14.1−2.49i)T+(62.9−22.9i)T2 1 + (14.1 - 2.49i)T + (62.9 - 22.9i)T^{2}
71 1+(7.01+5.88i)T+(12.3+69.9i)T2 1 + (7.01 + 5.88i)T + (12.3 + 69.9i)T^{2}
73 1+(−1.56+8.88i)T+(−68.5−24.9i)T2 1 + (-1.56 + 8.88i)T + (-68.5 - 24.9i)T^{2}
79 1+(6.78−8.09i)T+(−13.7−77.7i)T2 1 + (6.78 - 8.09i)T + (-13.7 - 77.7i)T^{2}
83 1−4.95T+83T2 1 - 4.95T + 83T^{2}
89 1+(−4.70+0.828i)T+(83.6−30.4i)T2 1 + (-4.70 + 0.828i)T + (83.6 - 30.4i)T^{2}
97 1+(−0.862+4.89i)T+(−91.1−33.1i)T2 1 + (-0.862 + 4.89i)T + (-91.1 - 33.1i)T^{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

−10.16291551323617372762406312956, −8.800671032852934774732997519344, −8.378120559197458419127753952965, −7.59554938217738697128464818381, −6.78085015323937676472710089579, −5.98818479812714652994364932109, −4.13188038450588338705338008414, −3.04030366601179285789682919801, −1.72064263710990473630800852689, −0.38404225513943878681392406001, 1.89660669370613944029766633281, 3.49747527743143775127589590638, 4.24708151675710696549407526816, 5.95171421360784305400209461318, 6.30245913239970774713273738060, 7.893017846088385129984111632456, 8.445050183209850125461294644531, 9.059875915340677064149769195204, 9.992045508058294715747635549809, 10.77107236626374366660849070832

Graph of the ZZ-function along the critical line