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Properties

Label 2-416-104.101-c1-0-0
Degree 22
Conductor 416416
Sign −0.859−0.510i-0.859 - 0.510i
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  + (−0.323 + 0.186i)3-s − 2.04·5-s + (0.768 + 0.443i)7-s + (−1.43 + 2.47i)9-s + (−0.654 − 1.13i)11-s + (−3.57 − 0.490i)13-s + (0.661 − 0.381i)15-s + (−1.02 + 1.76i)17-s + (−3.29 + 5.70i)19-s − 0.331·21-s + (−1.76 − 3.06i)23-s − 0.816·25-s − 2.18i·27-s + (−2.84 + 1.64i)29-s + 7.97i·31-s + ⋯
L(s)  = 1  + (−0.186 + 0.107i)3-s − 0.914·5-s + (0.290 + 0.167i)7-s + (−0.476 + 0.825i)9-s + (−0.197 − 0.341i)11-s + (−0.990 − 0.136i)13-s + (0.170 − 0.0985i)15-s + (−0.247 + 0.429i)17-s + (−0.755 + 1.30i)19-s − 0.0722·21-s + (−0.368 − 0.638i)23-s − 0.163·25-s − 0.421i·27-s + (−0.529 + 0.305i)29-s + 1.43i·31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.859−0.510i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.859−0.510i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.859−0.510i-0.859 - 0.510i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(49,⋅)\chi_{416} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.859−0.510i)(2,\ 416,\ (\ :1/2),\ -0.859 - 0.510i)

Particular Values

L(1)L(1) ≈\approx 0.107759+0.392189i0.107759 + 0.392189i
L(12)L(\frac12) ≈\approx 0.107759+0.392189i0.107759 + 0.392189i
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
13 1+(3.57+0.490i)T 1 + (3.57 + 0.490i)T
good3 1+(0.323−0.186i)T+(1.5−2.59i)T2 1 + (0.323 - 0.186i)T + (1.5 - 2.59i)T^{2}
5 1+2.04T+5T2 1 + 2.04T + 5T^{2}
7 1+(−0.768−0.443i)T+(3.5+6.06i)T2 1 + (-0.768 - 0.443i)T + (3.5 + 6.06i)T^{2}
11 1+(0.654+1.13i)T+(−5.5+9.52i)T2 1 + (0.654 + 1.13i)T + (-5.5 + 9.52i)T^{2}
17 1+(1.02−1.76i)T+(−8.5−14.7i)T2 1 + (1.02 - 1.76i)T + (-8.5 - 14.7i)T^{2}
19 1+(3.29−5.70i)T+(−9.5−16.4i)T2 1 + (3.29 - 5.70i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.76+3.06i)T+(−11.5+19.9i)T2 1 + (1.76 + 3.06i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.84−1.64i)T+(14.5−25.1i)T2 1 + (2.84 - 1.64i)T + (14.5 - 25.1i)T^{2}
31 1−7.97iT−31T2 1 - 7.97iT - 31T^{2}
37 1+(−2.25−3.90i)T+(−18.5+32.0i)T2 1 + (-2.25 - 3.90i)T + (-18.5 + 32.0i)T^{2}
41 1+(−6.06+3.50i)T+(20.5−35.5i)T2 1 + (-6.06 + 3.50i)T + (20.5 - 35.5i)T^{2}
43 1+(5.81+3.35i)T+(21.5+37.2i)T2 1 + (5.81 + 3.35i)T + (21.5 + 37.2i)T^{2}
47 1+0.0334iT−47T2 1 + 0.0334iT - 47T^{2}
53 1+2.79iT−53T2 1 + 2.79iT - 53T^{2}
59 1+(−3.94+6.82i)T+(−29.5−51.0i)T2 1 + (-3.94 + 6.82i)T + (-29.5 - 51.0i)T^{2}
61 1+(−11.4−6.62i)T+(30.5+52.8i)T2 1 + (-11.4 - 6.62i)T + (30.5 + 52.8i)T^{2}
67 1+(−6.31−10.9i)T+(−33.5+58.0i)T2 1 + (-6.31 - 10.9i)T + (-33.5 + 58.0i)T^{2}
71 1+(10.0+5.78i)T+(35.5+61.4i)T2 1 + (10.0 + 5.78i)T + (35.5 + 61.4i)T^{2}
73 1−3.99iT−73T2 1 - 3.99iT - 73T^{2}
79 1−1.04T+79T2 1 - 1.04T + 79T^{2}
83 1−11.8T+83T2 1 - 11.8T + 83T^{2}
89 1+(12.5−7.23i)T+(44.5−77.0i)T2 1 + (12.5 - 7.23i)T + (44.5 - 77.0i)T^{2}
97 1+(−0.592−0.342i)T+(48.5+84.0i)T2 1 + (-0.592 - 0.342i)T + (48.5 + 84.0i)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

−11.59537140038712339882458890162, −10.71636623408150912651963091469, −10.02442674387675930329928869876, −8.458809423556299394104348928595, −8.145705262691588985831853125588, −7.04817882451067019317527895100, −5.74606952235436261245102632512, −4.80519661238100177750558412846, −3.69311297514648904253740285431, −2.21058465006549091592255359985, 0.25209605739190198611799124885, 2.48863828395004027914805612209, 3.93019332247272793796256636687, 4.84372935517033923772994542881, 6.15044161945329785448288349171, 7.24675551433309809675193192692, 7.88738185788115738487558053368, 9.085457378090370431233272645683, 9.817903943992353817862354655628, 11.31408820654202236105727610653

Graph of the ZZ-function along the critical line