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Properties

Label 2-416-104.77-c1-0-11
Degree 22
Conductor 416416
Sign −0.937−0.347i-0.937 - 0.347i
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  − 2.94i·3-s − 1.81·5-s + 1.13i·7-s − 5.70·9-s − 4.40·11-s + (−2.58 − 2.50i)13-s + 5.35i·15-s + 0.701·17-s + 5.95·19-s + 3.36·21-s − 4·23-s − 1.70·25-s + 7.96i·27-s + 5.01i·29-s − 8.77i·31-s + ⋯
L(s)  = 1  − 1.70i·3-s − 0.812·5-s + 0.430i·7-s − 1.90·9-s − 1.32·11-s + (−0.717 − 0.696i)13-s + 1.38i·15-s + 0.170·17-s + 1.36·19-s + 0.733·21-s − 0.834·23-s − 0.340·25-s + 1.53i·27-s + 0.932i·29-s − 1.57i·31-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.937−0.347i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.937−0.347i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.937−0.347i-0.937 - 0.347i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(337,⋅)\chi_{416} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), −0.937−0.347i)(2,\ 416,\ (\ :1/2),\ -0.937 - 0.347i)

Particular Values

L(1)L(1) ≈\approx 0.0915884+0.510208i0.0915884 + 0.510208i
L(12)L(\frac12) ≈\approx 0.0915884+0.510208i0.0915884 + 0.510208i
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+(2.58+2.50i)T 1 + (2.58 + 2.50i)T
good3 1+2.94iT−3T2 1 + 2.94iT - 3T^{2}
5 1+1.81T+5T2 1 + 1.81T + 5T^{2}
7 1−1.13iT−7T2 1 - 1.13iT - 7T^{2}
11 1+4.40T+11T2 1 + 4.40T + 11T^{2}
17 1−0.701T+17T2 1 - 0.701T + 17T^{2}
19 1−5.95T+19T2 1 - 5.95T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−5.01iT−29T2 1 - 5.01iT - 29T^{2}
31 1+8.77iT−31T2 1 + 8.77iT - 31T^{2}
37 1+3.36T+37T2 1 + 3.36T + 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1+2.94iT−43T2 1 + 2.94iT - 43T^{2}
47 1−1.13iT−47T2 1 - 1.13iT - 47T^{2}
53 1+11.7iT−53T2 1 + 11.7iT - 53T^{2}
59 1+5.95T+59T2 1 + 5.95T + 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1+7.49T+67T2 1 + 7.49T + 67T^{2}
71 1+11.8iT−71T2 1 + 11.8iT - 71T^{2}
73 1+8.43iT−73T2 1 + 8.43iT - 73T^{2}
79 1−10.8T+79T2 1 - 10.8T + 79T^{2}
83 1+2.85T+83T2 1 + 2.85T + 83T^{2}
89 1−8.43iT−89T2 1 - 8.43iT - 89T^{2}
97 1+12.9iT−97T2 1 + 12.9iT - 97T^{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

−10.99178950585917456390087883236, −9.795732005704590118602089496325, −8.418973842480414703745405096928, −7.64176828531845958891868448990, −7.41213846336651088655333543357, −5.98815620957326193323621835520, −5.16052513303897782227303704355, −3.23351120880476578568853157210, −2.14780028193559796460141726380, −0.31544598020198545195161076507, 2.89520133806281409352524074924, 3.93996541154101015795917268034, 4.75462238730020055629205013888, 5.61105944559796117999197351782, 7.31300741327736845742177101069, 8.103132825183062001092502209776, 9.227934997177447201379396904028, 10.04883037980242445504051267279, 10.58957815903099972553793601997, 11.55493088385701449822211076504

Graph of the ZZ-function along the critical line