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Properties

Label 2-520-104.101-c1-0-14
Degree 22
Conductor 520520
Sign 0.0796−0.996i0.0796 - 0.996i
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  + (−1.30 + 0.554i)2-s + (0.176 − 0.101i)3-s + (1.38 − 1.44i)4-s + 5-s + (−0.173 + 0.230i)6-s + (0.820 + 0.473i)7-s + (−0.999 + 2.64i)8-s + (−1.47 + 2.56i)9-s + (−1.30 + 0.554i)10-s + (0.455 + 0.788i)11-s + (0.0972 − 0.395i)12-s + (−2.46 + 2.63i)13-s + (−1.33 − 0.160i)14-s + (0.176 − 0.101i)15-s + (−0.168 − 3.99i)16-s + (−1.02 + 1.77i)17-s + ⋯
L(s)  = 1  + (−0.919 + 0.392i)2-s + (0.101 − 0.0588i)3-s + (0.692 − 0.721i)4-s + 0.447·5-s + (−0.0706 + 0.0941i)6-s + (0.310 + 0.179i)7-s + (−0.353 + 0.935i)8-s + (−0.493 + 0.854i)9-s + (−0.411 + 0.175i)10-s + (0.137 + 0.237i)11-s + (0.0280 − 0.114i)12-s + (−0.683 + 0.729i)13-s + (−0.355 − 0.0429i)14-s + (0.0455 − 0.0263i)15-s + (−0.0421 − 0.999i)16-s + (−0.248 + 0.431i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((0.0796−0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((0.0796−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 0.0796−0.996i0.0796 - 0.996i
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), 0.0796−0.996i)(2,\ 520,\ (\ :1/2),\ 0.0796 - 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.699677+0.645986i0.699677 + 0.645986i
L(12)L(\frac12) ≈\approx 0.699677+0.645986i0.699677 + 0.645986i
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+(1.30−0.554i)T 1 + (1.30 - 0.554i)T
5 1−T 1 - T
13 1+(2.46−2.63i)T 1 + (2.46 - 2.63i)T
good3 1+(−0.176+0.101i)T+(1.5−2.59i)T2 1 + (-0.176 + 0.101i)T + (1.5 - 2.59i)T^{2}
7 1+(−0.820−0.473i)T+(3.5+6.06i)T2 1 + (-0.820 - 0.473i)T + (3.5 + 6.06i)T^{2}
11 1+(−0.455−0.788i)T+(−5.5+9.52i)T2 1 + (-0.455 - 0.788i)T + (-5.5 + 9.52i)T^{2}
17 1+(1.02−1.77i)T+(−8.5−14.7i)T2 1 + (1.02 - 1.77i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.975+1.68i)T+(−9.5−16.4i)T2 1 + (-0.975 + 1.68i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.80−6.58i)T+(−11.5+19.9i)T2 1 + (-3.80 - 6.58i)T + (-11.5 + 19.9i)T^{2}
29 1+(−7.35+4.24i)T+(14.5−25.1i)T2 1 + (-7.35 + 4.24i)T + (14.5 - 25.1i)T^{2}
31 1−6.07iT−31T2 1 - 6.07iT - 31T^{2}
37 1+(5.00+8.66i)T+(−18.5+32.0i)T2 1 + (5.00 + 8.66i)T + (-18.5 + 32.0i)T^{2}
41 1+(4.12−2.37i)T+(20.5−35.5i)T2 1 + (4.12 - 2.37i)T + (20.5 - 35.5i)T^{2}
43 1+(−4.14−2.39i)T+(21.5+37.2i)T2 1 + (-4.14 - 2.39i)T + (21.5 + 37.2i)T^{2}
47 1−10.4iT−47T2 1 - 10.4iT - 47T^{2}
53 1−11.4iT−53T2 1 - 11.4iT - 53T^{2}
59 1+(−4.67+8.09i)T+(−29.5−51.0i)T2 1 + (-4.67 + 8.09i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.13−1.81i)T+(30.5+52.8i)T2 1 + (-3.13 - 1.81i)T + (30.5 + 52.8i)T^{2}
67 1+(0.283+0.491i)T+(−33.5+58.0i)T2 1 + (0.283 + 0.491i)T + (-33.5 + 58.0i)T^{2}
71 1+(−8.63−4.98i)T+(35.5+61.4i)T2 1 + (-8.63 - 4.98i)T + (35.5 + 61.4i)T^{2}
73 1−1.67iT−73T2 1 - 1.67iT - 73T^{2}
79 1−5.13T+79T2 1 - 5.13T + 79T^{2}
83 1+15.0T+83T2 1 + 15.0T + 83T^{2}
89 1+(1.09−0.634i)T+(44.5−77.0i)T2 1 + (1.09 - 0.634i)T + (44.5 - 77.0i)T^{2}
97 1+(6.15+3.55i)T+(48.5+84.0i)T2 1 + (6.15 + 3.55i)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

−10.97882022457695584469151245992, −10.03510670539988173099648595143, −9.226414074617937549347954200620, −8.493361157099738673754227071196, −7.53381779246546820518224360462, −6.74974327836501840978314972653, −5.60627502144894919236160114379, −4.78002797585200968914669903065, −2.71248203388969820265342267143, −1.64541499479438974188194397674, 0.76470986271981354576136317840, 2.46404727609633376248121839187, 3.44264558712434289879259311658, 4.98813574211627326999586478497, 6.33589991779199070785204836393, 7.09915261038516645517193852894, 8.329237694881032459937925497375, 8.830864711139020381053759031108, 9.895599838529008787999698702445, 10.41527900862204072383422763766

Graph of the ZZ-function along the critical line