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Properties

Label 2-520-104.101-c1-0-12
Degree 22
Conductor 520520
Sign 0.798−0.602i0.798 - 0.602i
Analytic cond. 4.152224.15222
Root an. cond. 2.037692.03769
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  + (−0.936 − 1.05i)2-s + (0.700 − 0.404i)3-s + (−0.245 + 1.98i)4-s − 5-s + (−1.08 − 0.363i)6-s + (3.38 + 1.95i)7-s + (2.33 − 1.59i)8-s + (−1.17 + 2.03i)9-s + (0.936 + 1.05i)10-s + (−0.223 − 0.387i)11-s + (0.630 + 1.48i)12-s + (−3.01 + 1.98i)13-s + (−1.09 − 5.41i)14-s + (−0.700 + 0.404i)15-s + (−3.87 − 0.975i)16-s + (−4.00 + 6.94i)17-s + ⋯
L(s)  = 1  + (−0.662 − 0.749i)2-s + (0.404 − 0.233i)3-s + (−0.122 + 0.992i)4-s − 0.447·5-s + (−0.442 − 0.148i)6-s + (1.27 + 0.737i)7-s + (0.824 − 0.565i)8-s + (−0.390 + 0.677i)9-s + (0.296 + 0.335i)10-s + (−0.0674 − 0.116i)11-s + (0.182 + 0.429i)12-s + (−0.835 + 0.550i)13-s + (−0.293 − 1.44i)14-s + (−0.180 + 0.104i)15-s + (−0.969 − 0.243i)16-s + (−0.972 + 1.68i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((0.798−0.602i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((0.798−0.602i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 0.798−0.602i0.798 - 0.602i
Analytic conductor: 4.152224.15222
Root analytic conductor: 2.037692.03769
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ520(101,⋅)\chi_{520} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 520, ( :1/2), 0.798−0.602i)(2,\ 520,\ (\ :1/2),\ 0.798 - 0.602i)

Particular Values

L(1)L(1) ≈\approx 0.929456+0.311530i0.929456 + 0.311530i
L(12)L(\frac12) ≈\approx 0.929456+0.311530i0.929456 + 0.311530i
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+(0.936+1.05i)T 1 + (0.936 + 1.05i)T
5 1+T 1 + T
13 1+(3.01−1.98i)T 1 + (3.01 - 1.98i)T
good3 1+(−0.700+0.404i)T+(1.5−2.59i)T2 1 + (-0.700 + 0.404i)T + (1.5 - 2.59i)T^{2}
7 1+(−3.38−1.95i)T+(3.5+6.06i)T2 1 + (-3.38 - 1.95i)T + (3.5 + 6.06i)T^{2}
11 1+(0.223+0.387i)T+(−5.5+9.52i)T2 1 + (0.223 + 0.387i)T + (-5.5 + 9.52i)T^{2}
17 1+(4.00−6.94i)T+(−8.5−14.7i)T2 1 + (4.00 - 6.94i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.861−1.49i)T+(−9.5−16.4i)T2 1 + (0.861 - 1.49i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.570+0.987i)T+(−11.5+19.9i)T2 1 + (0.570 + 0.987i)T + (-11.5 + 19.9i)T^{2}
29 1+(−8.04+4.64i)T+(14.5−25.1i)T2 1 + (-8.04 + 4.64i)T + (14.5 - 25.1i)T^{2}
31 1−5.19iT−31T2 1 - 5.19iT - 31T^{2}
37 1+(−2.59−4.49i)T+(−18.5+32.0i)T2 1 + (-2.59 - 4.49i)T + (-18.5 + 32.0i)T^{2}
41 1+(−4.37+2.52i)T+(20.5−35.5i)T2 1 + (-4.37 + 2.52i)T + (20.5 - 35.5i)T^{2}
43 1+(−0.587−0.339i)T+(21.5+37.2i)T2 1 + (-0.587 - 0.339i)T + (21.5 + 37.2i)T^{2}
47 1+8.71iT−47T2 1 + 8.71iT - 47T^{2}
53 1+11.8iT−53T2 1 + 11.8iT - 53T^{2}
59 1+(4.67−8.08i)T+(−29.5−51.0i)T2 1 + (4.67 - 8.08i)T + (-29.5 - 51.0i)T^{2}
61 1+(6.84+3.95i)T+(30.5+52.8i)T2 1 + (6.84 + 3.95i)T + (30.5 + 52.8i)T^{2}
67 1+(−5.35−9.28i)T+(−33.5+58.0i)T2 1 + (-5.35 - 9.28i)T + (-33.5 + 58.0i)T^{2}
71 1+(−9.05−5.22i)T+(35.5+61.4i)T2 1 + (-9.05 - 5.22i)T + (35.5 + 61.4i)T^{2}
73 1+0.118iT−73T2 1 + 0.118iT - 73T^{2}
79 1−17.2T+79T2 1 - 17.2T + 79T^{2}
83 1+10.1T+83T2 1 + 10.1T + 83T^{2}
89 1+(−7.70+4.44i)T+(44.5−77.0i)T2 1 + (-7.70 + 4.44i)T + (44.5 - 77.0i)T^{2}
97 1+(−7.84−4.52i)T+(48.5+84.0i)T2 1 + (-7.84 - 4.52i)T + (48.5 + 84.0i)T^{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

−10.96678295177877004565976040554, −10.26139618414909694536028155055, −8.928446254991227496480295361799, −8.281415536984953503996108191136, −7.964334717197227269386762879328, −6.65935789110566449628648746342, −5.03710532009695412511461726179, −4.12293812148106245898870592775, −2.55173997710747352536748535804, −1.80113142926449682062276100195, 0.69494584163210744270133801393, 2.61260153065509351850543415118, 4.41973782787157175483890653159, 4.98867686768257298871893604743, 6.44234904459110498052870950549, 7.49190665356538568492936930104, 7.917767746418074018194618160653, 8.993965614441257258868738626768, 9.589197860613989273250270050643, 10.78357132284896250865813704853

Graph of the ZZ-function along the critical line