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Properties

Label 2-702-39.5-c1-0-7
Degree 22
Conductor 702702
Sign 0.747−0.663i0.747 - 0.663i
Analytic cond. 5.605495.60549
Root an. cond. 2.367592.36759
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−3.09 − 3.09i)5-s + (2.51 + 2.51i)7-s + (−0.707 + 0.707i)8-s − 4.37i·10-s + (−1.30 + 1.30i)11-s + (3.38 + 1.24i)13-s + 3.55i·14-s − 1.00·16-s + 7.96·17-s + (1.81 − 1.81i)19-s + (3.09 − 3.09i)20-s − 1.83·22-s + 4.57·23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−1.38 − 1.38i)5-s + (0.949 + 0.949i)7-s + (−0.250 + 0.250i)8-s − 1.38i·10-s + (−0.392 + 0.392i)11-s + (0.938 + 0.346i)13-s + 0.949i·14-s − 0.250·16-s + 1.93·17-s + (0.416 − 0.416i)19-s + (0.691 − 0.691i)20-s − 0.392·22-s + 0.954·23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((0.747−0.663i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((0.747−0.663i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: 0.747−0.663i0.747 - 0.663i
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ702(161,⋅)\chi_{702} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), 0.747−0.663i)(2,\ 702,\ (\ :1/2),\ 0.747 - 0.663i)

Particular Values

L(1)L(1) ≈\approx 1.67864+0.637661i1.67864 + 0.637661i
L(12)L(\frac12) ≈\approx 1.67864+0.637661i1.67864 + 0.637661i
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
3 1 1
13 1+(−3.38−1.24i)T 1 + (-3.38 - 1.24i)T
good5 1+(3.09+3.09i)T+5iT2 1 + (3.09 + 3.09i)T + 5iT^{2}
7 1+(−2.51−2.51i)T+7iT2 1 + (-2.51 - 2.51i)T + 7iT^{2}
11 1+(1.30−1.30i)T−11iT2 1 + (1.30 - 1.30i)T - 11iT^{2}
17 1−7.96T+17T2 1 - 7.96T + 17T^{2}
19 1+(−1.81+1.81i)T−19iT2 1 + (-1.81 + 1.81i)T - 19iT^{2}
23 1−4.57T+23T2 1 - 4.57T + 23T^{2}
29 1−2.20iT−29T2 1 - 2.20iT - 29T^{2}
31 1+(−1.85+1.85i)T−31iT2 1 + (-1.85 + 1.85i)T - 31iT^{2}
37 1+(2.39+2.39i)T+37iT2 1 + (2.39 + 2.39i)T + 37iT^{2}
41 1+(−2.35−2.35i)T+41iT2 1 + (-2.35 - 2.35i)T + 41iT^{2}
43 1+6.20iT−43T2 1 + 6.20iT - 43T^{2}
47 1+(−2.43+2.43i)T−47iT2 1 + (-2.43 + 2.43i)T - 47iT^{2}
53 1−8.05iT−53T2 1 - 8.05iT - 53T^{2}
59 1+(9.17−9.17i)T−59iT2 1 + (9.17 - 9.17i)T - 59iT^{2}
61 1+6.76T+61T2 1 + 6.76T + 61T^{2}
67 1+(−4.76+4.76i)T−67iT2 1 + (-4.76 + 4.76i)T - 67iT^{2}
71 1+(7.32+7.32i)T+71iT2 1 + (7.32 + 7.32i)T + 71iT^{2}
73 1+(−5.48−5.48i)T+73iT2 1 + (-5.48 - 5.48i)T + 73iT^{2}
79 1+2.87T+79T2 1 + 2.87T + 79T^{2}
83 1+(7.25+7.25i)T+83iT2 1 + (7.25 + 7.25i)T + 83iT^{2}
89 1+(4.33−4.33i)T−89iT2 1 + (4.33 - 4.33i)T - 89iT^{2}
97 1+(−3.13+3.13i)T−97iT2 1 + (-3.13 + 3.13i)T - 97iT^{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

−10.84435171668600726536660497513, −9.207407597859172427448906347298, −8.668053677060800628694880144542, −7.88821807974700050479189067562, −7.36644653427162994456551176264, −5.68398329049552762001538770342, −5.11220363009803444280547492375, −4.32152161497045346131237503377, −3.22576722722332490256273838404, −1.27545314820180322096595369679, 1.05923779109310341025892612717, 3.08810434064775349475369823404, 3.54577901483828060179872847920, 4.57113549261084725300211173364, 5.78791398857576704643187496074, 6.95483823895697588659170185347, 7.78502144662796945682162982802, 8.212906625189023733991287486904, 9.973600570292079481548119967149, 10.68418720853352429159956964571

Graph of the ZZ-function along the critical line