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Properties

Label 2-416-13.3-c1-0-10
Degree 22
Conductor 416416
Sign −0.0128+0.999i-0.0128 + 0.999i
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  + (0.207 + 0.358i)3-s − 2.82·5-s + (0.792 − 1.37i)7-s + (1.41 − 2.44i)9-s + (−2.62 − 4.54i)11-s + (−1 + 3.46i)13-s + (−0.585 − 1.01i)15-s + (0.0857 − 0.148i)17-s + (3.62 − 6.27i)19-s + 0.656·21-s + (−3.62 − 6.27i)23-s + 3.00·25-s + 2.41·27-s + (1.32 + 2.30i)29-s + 5.65·31-s + ⋯
L(s)  = 1  + (0.119 + 0.207i)3-s − 1.26·5-s + (0.299 − 0.519i)7-s + (0.471 − 0.816i)9-s + (−0.790 − 1.36i)11-s + (−0.277 + 0.960i)13-s + (−0.151 − 0.261i)15-s + (0.0208 − 0.0360i)17-s + (0.830 − 1.43i)19-s + 0.143·21-s + (−0.755 − 1.30i)23-s + 0.600·25-s + 0.464·27-s + (0.246 + 0.427i)29-s + 1.01·31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.0128+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.0128+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.0128+0.999i-0.0128 + 0.999i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(289,⋅)\chi_{416} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.0128+0.999i)(2,\ 416,\ (\ :1/2),\ -0.0128 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.649541−0.657924i0.649541 - 0.657924i
L(12)L(\frac12) ≈\approx 0.649541−0.657924i0.649541 - 0.657924i
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+(1−3.46i)T 1 + (1 - 3.46i)T
good3 1+(−0.207−0.358i)T+(−1.5+2.59i)T2 1 + (-0.207 - 0.358i)T + (-1.5 + 2.59i)T^{2}
5 1+2.82T+5T2 1 + 2.82T + 5T^{2}
7 1+(−0.792+1.37i)T+(−3.5−6.06i)T2 1 + (-0.792 + 1.37i)T + (-3.5 - 6.06i)T^{2}
11 1+(2.62+4.54i)T+(−5.5+9.52i)T2 1 + (2.62 + 4.54i)T + (-5.5 + 9.52i)T^{2}
17 1+(−0.0857+0.148i)T+(−8.5−14.7i)T2 1 + (-0.0857 + 0.148i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.62+6.27i)T+(−9.5−16.4i)T2 1 + (-3.62 + 6.27i)T + (-9.5 - 16.4i)T^{2}
23 1+(3.62+6.27i)T+(−11.5+19.9i)T2 1 + (3.62 + 6.27i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.32−2.30i)T+(−14.5+25.1i)T2 1 + (-1.32 - 2.30i)T + (-14.5 + 25.1i)T^{2}
31 1−5.65T+31T2 1 - 5.65T + 31T^{2}
37 1+(4.74+8.21i)T+(−18.5+32.0i)T2 1 + (4.74 + 8.21i)T + (-18.5 + 32.0i)T^{2}
41 1+(−0.0857−0.148i)T+(−20.5+35.5i)T2 1 + (-0.0857 - 0.148i)T + (-20.5 + 35.5i)T^{2}
43 1+(5.03−8.72i)T+(−21.5−37.2i)T2 1 + (5.03 - 8.72i)T + (-21.5 - 37.2i)T^{2}
47 1+6T+47T2 1 + 6T + 47T^{2}
53 1−2.82T+53T2 1 - 2.82T + 53T^{2}
59 1+(−3.62+6.27i)T+(−29.5−51.0i)T2 1 + (-3.62 + 6.27i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.5−6.06i)T+(−30.5−52.8i)T2 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(−2.37−4.11i)T+(−33.5+58.0i)T2 1 + (-2.37 - 4.11i)T + (-33.5 + 58.0i)T^{2}
71 1+(−0.621+1.07i)T+(−35.5−61.4i)T2 1 + (-0.621 + 1.07i)T + (-35.5 - 61.4i)T^{2}
73 1+4.48T+73T2 1 + 4.48T + 73T^{2}
79 1−6T+79T2 1 - 6T + 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1+(−7.32−12.6i)T+(−44.5+77.0i)T2 1 + (-7.32 - 12.6i)T + (-44.5 + 77.0i)T^{2}
97 1+(−4.5+7.79i)T+(−48.5−84.0i)T2 1 + (-4.5 + 7.79i)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

−11.14705572895428063448877417336, −10.18257573336889572028925933437, −9.020135412983367824107771367280, −8.242032035378495833313999001254, −7.32545419090055812618387222217, −6.44681822157245728736762298299, −4.85611813310988074459383364679, −4.02518619012058607441005479869, −2.97781434624446805719630255174, −0.59675671118218918460236937320, 1.92262402817580827586058454024, 3.39697408338966575749145793737, 4.66746644552499408139395255332, 5.48198528747405048092641515389, 7.16301982154919587188416679669, 7.890454310843357315933530154568, 8.196595566786757182725774877896, 9.950637336760670557166965531338, 10.32361202741907574121250582253, 11.83669854895216553203583891629

Graph of the ZZ-function along the critical line