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Properties

Label 2-520-104.69-c1-0-49
Degree 22
Conductor 520520
Sign −0.850+0.526i-0.850 + 0.526i
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  + (0.449 − 1.34i)2-s + (−0.700 − 0.404i)3-s + (−1.59 − 1.20i)4-s + 5-s + (−0.856 + 0.757i)6-s + (3.38 − 1.95i)7-s + (−2.33 + 1.59i)8-s + (−1.17 − 2.03i)9-s + (0.449 − 1.34i)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 + (1.09 + 3.84i)16-s + (−4.00 − 6.94i)17-s + ⋯
L(s)  = 1  + (0.317 − 0.948i)2-s + (−0.404 − 0.233i)3-s + (−0.798 − 0.602i)4-s + 0.447·5-s + (−0.349 + 0.309i)6-s + (1.27 − 0.737i)7-s + (−0.824 + 0.565i)8-s + (−0.390 − 0.677i)9-s + (0.142 − 0.424i)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 + (0.273 + 0.961i)16-s + (−0.972 − 1.68i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((−0.850+0.526i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((−0.850+0.526i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: −0.850+0.526i-0.850 + 0.526i
Analytic conductor: 4.152224.15222
Root analytic conductor: 2.037692.03769
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ520(381,⋅)\chi_{520} (381, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 520, ( :1/2), −0.850+0.526i)(2,\ 520,\ (\ :1/2),\ -0.850 + 0.526i)

Particular Values

L(1)L(1) ≈\approx 0.406268−1.42736i0.406268 - 1.42736i
L(12)L(\frac12) ≈\approx 0.406268−1.42736i0.406268 - 1.42736i
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+(−0.449+1.34i)T 1 + (-0.449 + 1.34i)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

−10.94671515376808560644501072658, −9.661566666385859381020768194529, −9.053263850280276473761200449926, −7.916168206851739271119249442882, −6.64633131884868080016245209106, −5.64016587718895696627965118191, −4.68601647704830538481444674444, −3.69686841973136872811935162061, −2.13706024994443305806635067001, −0.886018413335328616178113363619, 2.03841056984391073905833425911, 3.83180120909163815780607157285, 5.04988890038525012336811849780, 5.54067718806699804390222925432, 6.42015801392699476507963958212, 7.72223585715391145386775750754, 8.527325990256431158173273499073, 9.021851038859528271989053435685, 10.60450468880456106804538680651, 11.06392982184148163950531488731

Graph of the ZZ-function along the critical line