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Properties

Label 2-416-416.99-c1-0-34
Degree 22
Conductor 416416
Sign 0.194+0.980i0.194 + 0.980i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
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  + (−1.29 − 0.572i)2-s + (0.174 − 0.420i)3-s + (1.34 + 1.47i)4-s + (1.36 − 3.30i)5-s + (−0.466 + 0.444i)6-s + 1.57·7-s + (−0.893 − 2.68i)8-s + (1.97 + 1.97i)9-s + (−3.66 + 3.49i)10-s + (1.42 + 0.591i)11-s + (0.857 − 0.308i)12-s + (1.31 − 3.35i)13-s + (−2.03 − 0.901i)14-s + (−1.15 − 1.15i)15-s + (−0.380 + 3.98i)16-s − 3.74·17-s + ⋯
L(s)  = 1  + (−0.914 − 0.404i)2-s + (0.100 − 0.242i)3-s + (0.672 + 0.739i)4-s + (0.612 − 1.47i)5-s + (−0.190 + 0.181i)6-s + 0.595·7-s + (−0.315 − 0.948i)8-s + (0.658 + 0.658i)9-s + (−1.15 + 1.10i)10-s + (0.430 + 0.178i)11-s + (0.247 − 0.0889i)12-s + (0.364 − 0.931i)13-s + (−0.544 − 0.240i)14-s + (−0.297 − 0.297i)15-s + (−0.0950 + 0.995i)16-s − 0.908·17-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((0.194+0.980i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((0.194+0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 0.194+0.980i0.194 + 0.980i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(99,⋅)\chi_{416} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 0.194+0.980i)(2,\ 416,\ (\ :1/2),\ 0.194 + 0.980i)

Particular Values

L(1)L(1) ≈\approx 0.912266−0.749503i0.912266 - 0.749503i
L(12)L(\frac12) ≈\approx 0.912266−0.749503i0.912266 - 0.749503i
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+(1.29+0.572i)T 1 + (1.29 + 0.572i)T
13 1+(−1.31+3.35i)T 1 + (-1.31 + 3.35i)T
good3 1+(−0.174+0.420i)T+(−2.12−2.12i)T2 1 + (-0.174 + 0.420i)T + (-2.12 - 2.12i)T^{2}
5 1+(−1.36+3.30i)T+(−3.53−3.53i)T2 1 + (-1.36 + 3.30i)T + (-3.53 - 3.53i)T^{2}
7 1−1.57T+7T2 1 - 1.57T + 7T^{2}
11 1+(−1.42−0.591i)T+(7.77+7.77i)T2 1 + (-1.42 - 0.591i)T + (7.77 + 7.77i)T^{2}
17 1+3.74T+17T2 1 + 3.74T + 17T^{2}
19 1+(−3.19−7.70i)T+(−13.4+13.4i)T2 1 + (-3.19 - 7.70i)T + (-13.4 + 13.4i)T^{2}
23 1+(3.88+3.88i)T+23iT2 1 + (3.88 + 3.88i)T + 23iT^{2}
29 1+(−3.22−1.33i)T+(20.5+20.5i)T2 1 + (-3.22 - 1.33i)T + (20.5 + 20.5i)T^{2}
31 1+(−1.41+1.41i)T−31iT2 1 + (-1.41 + 1.41i)T - 31iT^{2}
37 1+(1.01+0.418i)T+(26.1+26.1i)T2 1 + (1.01 + 0.418i)T + (26.1 + 26.1i)T^{2}
41 1+10.5iT−41T2 1 + 10.5iT - 41T^{2}
43 1+(0.478−0.198i)T+(30.4−30.4i)T2 1 + (0.478 - 0.198i)T + (30.4 - 30.4i)T^{2}
47 1+(1.47−1.47i)T−47iT2 1 + (1.47 - 1.47i)T - 47iT^{2}
53 1+(4.69−1.94i)T+(37.4−37.4i)T2 1 + (4.69 - 1.94i)T + (37.4 - 37.4i)T^{2}
59 1+(−2.03+4.91i)T+(−41.7−41.7i)T2 1 + (-2.03 + 4.91i)T + (-41.7 - 41.7i)T^{2}
61 1+(6.76+2.80i)T+(43.1+43.1i)T2 1 + (6.76 + 2.80i)T + (43.1 + 43.1i)T^{2}
67 1+(5.79−2.40i)T+(47.3−47.3i)T2 1 + (5.79 - 2.40i)T + (47.3 - 47.3i)T^{2}
71 1−5.97iT−71T2 1 - 5.97iT - 71T^{2}
73 1−1.85T+73T2 1 - 1.85T + 73T^{2}
79 1−12.4T+79T2 1 - 12.4T + 79T^{2}
83 1+(−4.57−11.0i)T+(−58.6+58.6i)T2 1 + (-4.57 - 11.0i)T + (-58.6 + 58.6i)T^{2}
89 1−13.4iT−89T2 1 - 13.4iT - 89T^{2}
97 1+(−3.24−3.24i)T+97iT2 1 + (-3.24 - 3.24i)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

−10.75831902042478367561461674736, −10.06295342010941862131969277578, −9.174539973517747352269681017196, −8.232222102657673474790338451709, −7.85831989102969014407659045764, −6.39036308384731141799716086350, −5.17795163453634087013946263402, −4.01817800644349294277184549620, −2.06315976226253288932293429825, −1.19387107390197202969420997085, 1.69491923218093880099836660849, 3.03435180236402031710270399137, 4.66920229922147613012232930768, 6.30109727936244895579977511628, 6.67683493975272077593229297155, 7.57338637556831100428449329151, 8.927344484746242294946723252071, 9.540666672059606294819202743477, 10.34692147442058192033978621603, 11.30235943110030309734721186895

Graph of the ZZ-function along the critical line