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Properties

Label 2-520-104.101-c1-0-27
Degree 22
Conductor 520520
Sign 0.914+0.404i0.914 + 0.404i
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  + (0.491 + 1.32i)2-s + (−2.26 + 1.30i)3-s + (−1.51 + 1.30i)4-s − 5-s + (−2.84 − 2.35i)6-s + (−3.24 − 1.87i)7-s + (−2.47 − 1.37i)8-s + (1.91 − 3.31i)9-s + (−0.491 − 1.32i)10-s + (2.13 + 3.69i)11-s + (1.72 − 4.93i)12-s + (2.92 − 2.11i)13-s + (0.889 − 5.22i)14-s + (2.26 − 1.30i)15-s + (0.600 − 3.95i)16-s + (0.829 − 1.43i)17-s + ⋯
L(s)  = 1  + (0.347 + 0.937i)2-s + (−1.30 + 0.754i)3-s + (−0.758 + 0.651i)4-s − 0.447·5-s + (−1.16 − 0.962i)6-s + (−1.22 − 0.708i)7-s + (−0.874 − 0.484i)8-s + (0.637 − 1.10i)9-s + (−0.155 − 0.419i)10-s + (0.643 + 1.11i)11-s + (0.499 − 1.42i)12-s + (0.810 − 0.585i)13-s + (0.237 − 1.39i)14-s + (0.584 − 0.337i)15-s + (0.150 − 0.988i)16-s + (0.201 − 0.348i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((0.914+0.404i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((0.914+0.404i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 0.914+0.404i0.914 + 0.404i
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), 0.914+0.404i)(2,\ 520,\ (\ :1/2),\ 0.914 + 0.404i)

Particular Values

L(1)L(1) ≈\approx 0.270927−0.0571983i0.270927 - 0.0571983i
L(12)L(\frac12) ≈\approx 0.270927−0.0571983i0.270927 - 0.0571983i
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+(−0.491−1.32i)T 1 + (-0.491 - 1.32i)T
5 1+T 1 + T
13 1+(−2.92+2.11i)T 1 + (-2.92 + 2.11i)T
good3 1+(2.26−1.30i)T+(1.5−2.59i)T2 1 + (2.26 - 1.30i)T + (1.5 - 2.59i)T^{2}
7 1+(3.24+1.87i)T+(3.5+6.06i)T2 1 + (3.24 + 1.87i)T + (3.5 + 6.06i)T^{2}
11 1+(−2.13−3.69i)T+(−5.5+9.52i)T2 1 + (-2.13 - 3.69i)T + (-5.5 + 9.52i)T^{2}
17 1+(−0.829+1.43i)T+(−8.5−14.7i)T2 1 + (-0.829 + 1.43i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.21−3.83i)T+(−9.5−16.4i)T2 1 + (2.21 - 3.83i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.329+0.570i)T+(−11.5+19.9i)T2 1 + (0.329 + 0.570i)T + (-11.5 + 19.9i)T^{2}
29 1+(6.59−3.80i)T+(14.5−25.1i)T2 1 + (6.59 - 3.80i)T + (14.5 - 25.1i)T^{2}
31 1+8.51iT−31T2 1 + 8.51iT - 31T^{2}
37 1+(3.03+5.25i)T+(−18.5+32.0i)T2 1 + (3.03 + 5.25i)T + (-18.5 + 32.0i)T^{2}
41 1+(−7.02+4.05i)T+(20.5−35.5i)T2 1 + (-7.02 + 4.05i)T + (20.5 - 35.5i)T^{2}
43 1+(3.74+2.16i)T+(21.5+37.2i)T2 1 + (3.74 + 2.16i)T + (21.5 + 37.2i)T^{2}
47 1−6.33iT−47T2 1 - 6.33iT - 47T^{2}
53 1+12.5iT−53T2 1 + 12.5iT - 53T^{2}
59 1+(−4.08+7.07i)T+(−29.5−51.0i)T2 1 + (-4.08 + 7.07i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.45−1.41i)T+(30.5+52.8i)T2 1 + (-2.45 - 1.41i)T + (30.5 + 52.8i)T^{2}
67 1+(3.38+5.86i)T+(−33.5+58.0i)T2 1 + (3.38 + 5.86i)T + (-33.5 + 58.0i)T^{2}
71 1+(3.25+1.87i)T+(35.5+61.4i)T2 1 + (3.25 + 1.87i)T + (35.5 + 61.4i)T^{2}
73 1+4.75iT−73T2 1 + 4.75iT - 73T^{2}
79 1+3.88T+79T2 1 + 3.88T + 79T^{2}
83 1+3.60T+83T2 1 + 3.60T + 83T^{2}
89 1+(6.57−3.79i)T+(44.5−77.0i)T2 1 + (6.57 - 3.79i)T + (44.5 - 77.0i)T^{2}
97 1+(0.424+0.244i)T+(48.5+84.0i)T2 1 + (0.424 + 0.244i)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

−10.74746548829274372039743391564, −9.897979415075370213558539921586, −9.241419599978357716371817810356, −7.80242007164381798885876443450, −6.88973116957410530043458809246, −6.16611311652629933255689206290, −5.33287704853816216501523321200, −4.09485507984778017940535989446, −3.71112327461693255035276046708, −0.19551577413291201373195538252, 1.24932579655261928956693571793, 2.99656649774990977460598628785, 4.08552340909324983813026820419, 5.52751504494864602222434963045, 6.15943824848632883178961465674, 6.82143840202988316405241541161, 8.563220874780506392708111919775, 9.222931053213402748543393042778, 10.44891749266350213027197319921, 11.30437924620381059913433608717

Graph of the ZZ-function along the critical line