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Properties

Label 2-28e2-16.5-c1-0-75
Degree 22
Conductor 784784
Sign −0.540+0.841i-0.540 + 0.841i
Analytic cond. 6.260276.26027
Root an. cond. 2.502052.50205
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  + (1.32 − 0.482i)2-s + (1.83 − 1.83i)3-s + (1.53 − 1.28i)4-s + (−2.13 − 2.13i)5-s + (1.55 − 3.33i)6-s + (1.42 − 2.44i)8-s − 3.75i·9-s + (−3.86 − 1.80i)10-s + (2.28 + 2.28i)11-s + (0.462 − 5.17i)12-s + (−2.52 + 2.52i)13-s − 7.83·15-s + (0.708 − 3.93i)16-s + 0.402·17-s + (−1.81 − 4.99i)18-s + (1.01 − 1.01i)19-s + ⋯
L(s)  = 1  + (0.939 − 0.341i)2-s + (1.06 − 1.06i)3-s + (0.767 − 0.641i)4-s + (−0.953 − 0.953i)5-s + (0.635 − 1.35i)6-s + (0.502 − 0.864i)8-s − 1.25i·9-s + (−1.22 − 0.570i)10-s + (0.690 + 0.690i)11-s + (0.133 − 1.49i)12-s + (−0.699 + 0.699i)13-s − 2.02·15-s + (0.177 − 0.984i)16-s + 0.0975·17-s + (−0.427 − 1.17i)18-s + (0.233 − 0.233i)19-s + ⋯

Functional equation

Λ(s)=(784s/2 ΓC(s) L(s)=((−0.540+0.841i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(784s/2 ΓC(s+1/2) L(s)=((−0.540+0.841i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 784784    =    24⋅722^{4} \cdot 7^{2}
Sign: −0.540+0.841i-0.540 + 0.841i
Analytic conductor: 6.260276.26027
Root analytic conductor: 2.502052.50205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ784(197,⋅)\chi_{784} (197, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 784, ( :1/2), −0.540+0.841i)(2,\ 784,\ (\ :1/2),\ -0.540 + 0.841i)

Particular Values

L(1)L(1) ≈\approx 1.59096−2.91217i1.59096 - 2.91217i
L(12)L(\frac12) ≈\approx 1.59096−2.91217i1.59096 - 2.91217i
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+(−1.32+0.482i)T 1 + (-1.32 + 0.482i)T
7 1 1
good3 1+(−1.83+1.83i)T−3iT2 1 + (-1.83 + 1.83i)T - 3iT^{2}
5 1+(2.13+2.13i)T+5iT2 1 + (2.13 + 2.13i)T + 5iT^{2}
11 1+(−2.28−2.28i)T+11iT2 1 + (-2.28 - 2.28i)T + 11iT^{2}
13 1+(2.52−2.52i)T−13iT2 1 + (2.52 - 2.52i)T - 13iT^{2}
17 1−0.402T+17T2 1 - 0.402T + 17T^{2}
19 1+(−1.01+1.01i)T−19iT2 1 + (-1.01 + 1.01i)T - 19iT^{2}
23 1−9.11iT−23T2 1 - 9.11iT - 23T^{2}
29 1+(−1.47+1.47i)T−29iT2 1 + (-1.47 + 1.47i)T - 29iT^{2}
31 1−4.25T+31T2 1 - 4.25T + 31T^{2}
37 1+(1.42+1.42i)T+37iT2 1 + (1.42 + 1.42i)T + 37iT^{2}
41 1+8.96iT−41T2 1 + 8.96iT - 41T^{2}
43 1+(0.997+0.997i)T+43iT2 1 + (0.997 + 0.997i)T + 43iT^{2}
47 1+4.19T+47T2 1 + 4.19T + 47T^{2}
53 1+(−1.33−1.33i)T+53iT2 1 + (-1.33 - 1.33i)T + 53iT^{2}
59 1+(−1.73−1.73i)T+59iT2 1 + (-1.73 - 1.73i)T + 59iT^{2}
61 1+(−1.87+1.87i)T−61iT2 1 + (-1.87 + 1.87i)T - 61iT^{2}
67 1+(−8.52+8.52i)T−67iT2 1 + (-8.52 + 8.52i)T - 67iT^{2}
71 1−0.451iT−71T2 1 - 0.451iT - 71T^{2}
73 1−10.8iT−73T2 1 - 10.8iT - 73T^{2}
79 1+12.6T+79T2 1 + 12.6T + 79T^{2}
83 1+(0.742−0.742i)T−83iT2 1 + (0.742 - 0.742i)T - 83iT^{2}
89 1−12.8iT−89T2 1 - 12.8iT - 89T^{2}
97 1+13.3T+97T2 1 + 13.3T + 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

−9.812002881535862721476051008974, −9.126134625687456060174018044411, −8.107014618180614624813859832476, −7.31811810680231637388258851935, −6.79090688996764983473111111596, −5.32139645113922608019443791373, −4.32107322725514960766560724478, −3.51456511712761894587876356498, −2.22445075207103795057863416559, −1.23306448981155822643854612686, 2.74716087880358944818843019461, 3.20169320748789079154055151625, 4.08834883906250503780467405923, 4.84086201562813928620793684404, 6.24448649349367700220000077318, 7.10338029116576209741209561714, 8.131833491578275057175882939694, 8.525149932022557596885005094717, 9.904137695547862169013681848665, 10.59206841652017572890590462278

Graph of the ZZ-function along the critical line