Properties

Label 2-776-776.539-c0-0-0
Degree 22
Conductor 776776
Sign 0.9880.150i-0.988 - 0.150i
Analytic cond. 0.3872740.387274
Root an. cond. 0.6223130.622313
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.5 + 0.133i)3-s + (−0.866 + 0.499i)4-s + (−0.258 − 0.448i)6-s + (−0.707 − 0.707i)8-s + (−0.633 + 0.366i)9-s + (−0.448 + 1.67i)11-s + (0.366 − 0.366i)12-s + (0.500 − 0.866i)16-s + (0.607 + 0.465i)17-s + (−0.517 − 0.517i)18-s + (−1.70 + 0.707i)19-s − 1.73·22-s + (0.448 + 0.258i)24-s + (−0.965 − 0.258i)25-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.5 + 0.133i)3-s + (−0.866 + 0.499i)4-s + (−0.258 − 0.448i)6-s + (−0.707 − 0.707i)8-s + (−0.633 + 0.366i)9-s + (−0.448 + 1.67i)11-s + (0.366 − 0.366i)12-s + (0.500 − 0.866i)16-s + (0.607 + 0.465i)17-s + (−0.517 − 0.517i)18-s + (−1.70 + 0.707i)19-s − 1.73·22-s + (0.448 + 0.258i)24-s + (−0.965 − 0.258i)25-s + ⋯

Functional equation

Λ(s)=(776s/2ΓC(s)L(s)=((0.9880.150i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(776s/2ΓC(s)L(s)=((0.9880.150i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 776776    =    23972^{3} \cdot 97
Sign: 0.9880.150i-0.988 - 0.150i
Analytic conductor: 0.3872740.387274
Root analytic conductor: 0.6223130.622313
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ776(539,)\chi_{776} (539, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 776, ( :0), 0.9880.150i)(2,\ 776,\ (\ :0),\ -0.988 - 0.150i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.61593033630.6159303363
L(12)L(\frac12) \approx 0.61593033630.6159303363
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.2580.965i)T 1 + (-0.258 - 0.965i)T
97 1+(0.965+0.258i)T 1 + (-0.965 + 0.258i)T
good3 1+(0.50.133i)T+(0.8660.5i)T2 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2}
5 1+(0.965+0.258i)T2 1 + (0.965 + 0.258i)T^{2}
7 1+(0.258+0.965i)T2 1 + (-0.258 + 0.965i)T^{2}
11 1+(0.4481.67i)T+(0.8660.5i)T2 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2}
13 1+(0.965+0.258i)T2 1 + (0.965 + 0.258i)T^{2}
17 1+(0.6070.465i)T+(0.258+0.965i)T2 1 + (-0.607 - 0.465i)T + (0.258 + 0.965i)T^{2}
19 1+(1.700.707i)T+(0.7070.707i)T2 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2}
23 1+(0.258+0.965i)T2 1 + (0.258 + 0.965i)T^{2}
29 1+(0.9650.258i)T2 1 + (-0.965 - 0.258i)T^{2}
31 1+(0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
37 1+(0.2580.965i)T2 1 + (0.258 - 0.965i)T^{2}
41 1+(1.830.241i)T+(0.965+0.258i)T2 1 + (-1.83 - 0.241i)T + (0.965 + 0.258i)T^{2}
43 1+(1.670.965i)T+(0.5+0.866i)T2 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2}
47 1+T2 1 + T^{2}
53 1+(0.8660.5i)T2 1 + (0.866 - 0.5i)T^{2}
59 1+(0.465+0.607i)T+(0.258+0.965i)T2 1 + (0.465 + 0.607i)T + (-0.258 + 0.965i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(1.830.758i)T+(0.7070.707i)T2 1 + (1.83 - 0.758i)T + (0.707 - 0.707i)T^{2}
71 1+(0.965+0.258i)T2 1 + (-0.965 + 0.258i)T^{2}
73 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
79 1iT2 1 - iT^{2}
83 1+(0.9650.741i)T+(0.258+0.965i)T2 1 + (-0.965 - 0.741i)T + (0.258 + 0.965i)T^{2}
89 1+(1.361.36i)T+iT2 1 + (-1.36 - 1.36i)T + iT^{2}
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   L(s)=p j=12(1αj,pps)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

−10.73818409727594933778739849942, −10.05088531229595852296275932457, −9.126815958579719719589482102132, −8.037831727905725490350999759741, −7.57285033966932380270249165507, −6.35275262755267581849566263579, −5.76689320552756515461892229368, −4.72744558259182194716843405847, −4.03481393650677739636832972219, −2.34931219759139854850813734566, 0.61880480843703651093382163743, 2.48455882516805052692740620302, 3.42335981777669448746719898302, 4.55781201495843219272400564559, 5.87058753213018978890023400031, 5.97913435975849543813228924201, 7.67220622493348100173052151093, 8.765967120107860215228710268017, 9.189515313106483178959743360759, 10.60615728948549847803231219739

Graph of the ZZ-function along the critical line