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Properties

Label 2-28e2-16.13-c1-0-14
Degree 22
Conductor 784784
Sign −0.577−0.816i-0.577 - 0.816i
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  + (−1.33 − 0.467i)2-s + (2.13 + 2.13i)3-s + (1.56 + 1.24i)4-s + (−0.545 + 0.545i)5-s + (−1.85 − 3.84i)6-s + (−1.50 − 2.39i)8-s + 6.10i·9-s + (0.983 − 0.473i)10-s + (−0.910 + 0.910i)11-s + (0.670 + 5.99i)12-s + (0.919 + 0.919i)13-s − 2.32·15-s + (0.883 + 3.90i)16-s − 7.91·17-s + (2.85 − 8.15i)18-s + (−1.30 − 1.30i)19-s + ⋯
L(s)  = 1  + (−0.943 − 0.330i)2-s + (1.23 + 1.23i)3-s + (0.781 + 0.624i)4-s + (−0.244 + 0.244i)5-s + (−0.755 − 1.57i)6-s + (−0.530 − 0.847i)8-s + 2.03i·9-s + (0.311 − 0.149i)10-s + (−0.274 + 0.274i)11-s + (0.193 + 1.73i)12-s + (0.254 + 0.254i)13-s − 0.601·15-s + (0.220 + 0.975i)16-s − 1.91·17-s + (0.673 − 1.92i)18-s + (−0.299 − 0.299i)19-s + ⋯

Functional equation

Λ(s)=(784s/2 ΓC(s) L(s)=((−0.577−0.816i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(784s/2 ΓC(s+1/2) L(s)=((−0.577−0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 784784    =    24⋅722^{4} \cdot 7^{2}
Sign: −0.577−0.816i-0.577 - 0.816i
Analytic conductor: 6.260276.26027
Root analytic conductor: 2.502052.50205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ784(589,⋅)\chi_{784} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 784, ( :1/2), −0.577−0.816i)(2,\ 784,\ (\ :1/2),\ -0.577 - 0.816i)

Particular Values

L(1)L(1) ≈\approx 0.550700+1.06370i0.550700 + 1.06370i
L(12)L(\frac12) ≈\approx 0.550700+1.06370i0.550700 + 1.06370i
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+(1.33+0.467i)T 1 + (1.33 + 0.467i)T
7 1 1
good3 1+(−2.13−2.13i)T+3iT2 1 + (-2.13 - 2.13i)T + 3iT^{2}
5 1+(0.545−0.545i)T−5iT2 1 + (0.545 - 0.545i)T - 5iT^{2}
11 1+(0.910−0.910i)T−11iT2 1 + (0.910 - 0.910i)T - 11iT^{2}
13 1+(−0.919−0.919i)T+13iT2 1 + (-0.919 - 0.919i)T + 13iT^{2}
17 1+7.91T+17T2 1 + 7.91T + 17T^{2}
19 1+(1.30+1.30i)T+19iT2 1 + (1.30 + 1.30i)T + 19iT^{2}
23 1−3.84iT−23T2 1 - 3.84iT - 23T^{2}
29 1+(−5.25−5.25i)T+29iT2 1 + (-5.25 - 5.25i)T + 29iT^{2}
31 1−4.89T+31T2 1 - 4.89T + 31T^{2}
37 1+(−0.938+0.938i)T−37iT2 1 + (-0.938 + 0.938i)T - 37iT^{2}
41 1+2.84iT−41T2 1 + 2.84iT - 41T^{2}
43 1+(−0.585+0.585i)T−43iT2 1 + (-0.585 + 0.585i)T - 43iT^{2}
47 1+5.72T+47T2 1 + 5.72T + 47T^{2}
53 1+(−6.96+6.96i)T−53iT2 1 + (-6.96 + 6.96i)T - 53iT^{2}
59 1+(6.54−6.54i)T−59iT2 1 + (6.54 - 6.54i)T - 59iT^{2}
61 1+(−4.67−4.67i)T+61iT2 1 + (-4.67 - 4.67i)T + 61iT^{2}
67 1+(4.35+4.35i)T+67iT2 1 + (4.35 + 4.35i)T + 67iT^{2}
71 1+1.99iT−71T2 1 + 1.99iT - 71T^{2}
73 1+7.72iT−73T2 1 + 7.72iT - 73T^{2}
79 1−9.26T+79T2 1 - 9.26T + 79T^{2}
83 1+(−4.78−4.78i)T+83iT2 1 + (-4.78 - 4.78i)T + 83iT^{2}
89 1−2.12iT−89T2 1 - 2.12iT - 89T^{2}
97 1−9.01T+97T2 1 - 9.01T + 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

−10.50775713261980698334753984917, −9.579067401116819627816904221423, −8.952083629133234518320718231133, −8.445031605354869598420004466894, −7.48293026265162347004076660278, −6.58199212977413954051131288259, −4.84056940552821318722982230212, −3.90265564132377356151532708550, −2.99379502581139051210507512252, −2.05605993736963304022880806319, 0.67258259938576826624874843171, 2.09802203029153356298591793645, 2.83200063952163528762097457172, 4.46789171899606956580474149554, 6.26389508735393068404164603804, 6.63538840990902271206206686424, 7.70861766300730112049528980492, 8.448746249280383512217780492058, 8.613279289711821579454008689942, 9.702286283817213184353900262205

Graph of the ZZ-function along the critical line