↑

Properties

Label 2-520-104.101-c1-0-13
Degree 22
Conductor 520520
Sign 0.972−0.233i0.972 - 0.233i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.615 − 1.27i)2-s + (−0.127 + 0.0734i)3-s + (−1.24 + 1.56i)4-s + 5-s + (0.171 + 0.116i)6-s + (−2.93 − 1.69i)7-s + (2.76 + 0.617i)8-s + (−1.48 + 2.57i)9-s + (−0.615 − 1.27i)10-s + (2.72 + 4.72i)11-s + (0.0429 − 0.290i)12-s + (1.96 − 3.01i)13-s + (−0.351 + 4.78i)14-s + (−0.127 + 0.0734i)15-s + (−0.911 − 3.89i)16-s + (−0.605 + 1.04i)17-s + ⋯
L(s)  = 1  + (−0.435 − 0.900i)2-s + (−0.0734 + 0.0423i)3-s + (−0.621 + 0.783i)4-s + 0.447·5-s + (0.0701 + 0.0476i)6-s + (−1.11 − 0.640i)7-s + (0.975 + 0.218i)8-s + (−0.496 + 0.859i)9-s + (−0.194 − 0.402i)10-s + (0.821 + 1.42i)11-s + (0.0124 − 0.0838i)12-s + (0.546 − 0.837i)13-s + (−0.0940 + 1.27i)14-s + (−0.0328 + 0.0189i)15-s + (−0.227 − 0.973i)16-s + (−0.146 + 0.254i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((0.972−0.233i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((0.972−0.233i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 0.972−0.233i0.972 - 0.233i
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), 0.972−0.233i)(2,\ 520,\ (\ :1/2),\ 0.972 - 0.233i)

Particular Values

L(1)L(1) ≈\approx 0.919689+0.109110i0.919689 + 0.109110i
L(12)L(\frac12) ≈\approx 0.919689+0.109110i0.919689 + 0.109110i
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+(0.615+1.27i)T 1 + (0.615 + 1.27i)T
5 1−T 1 - T
13 1+(−1.96+3.01i)T 1 + (-1.96 + 3.01i)T
good3 1+(0.127−0.0734i)T+(1.5−2.59i)T2 1 + (0.127 - 0.0734i)T + (1.5 - 2.59i)T^{2}
7 1+(2.93+1.69i)T+(3.5+6.06i)T2 1 + (2.93 + 1.69i)T + (3.5 + 6.06i)T^{2}
11 1+(−2.72−4.72i)T+(−5.5+9.52i)T2 1 + (-2.72 - 4.72i)T + (-5.5 + 9.52i)T^{2}
17 1+(0.605−1.04i)T+(−8.5−14.7i)T2 1 + (0.605 - 1.04i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.73−3.00i)T+(−9.5−16.4i)T2 1 + (1.73 - 3.00i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.76−6.52i)T+(−11.5+19.9i)T2 1 + (-3.76 - 6.52i)T + (-11.5 + 19.9i)T^{2}
29 1+(−6.10+3.52i)T+(14.5−25.1i)T2 1 + (-6.10 + 3.52i)T + (14.5 - 25.1i)T^{2}
31 1−7.60iT−31T2 1 - 7.60iT - 31T^{2}
37 1+(−3.82−6.62i)T+(−18.5+32.0i)T2 1 + (-3.82 - 6.62i)T + (-18.5 + 32.0i)T^{2}
41 1+(−3.47+2.00i)T+(20.5−35.5i)T2 1 + (-3.47 + 2.00i)T + (20.5 - 35.5i)T^{2}
43 1+(1.45+0.841i)T+(21.5+37.2i)T2 1 + (1.45 + 0.841i)T + (21.5 + 37.2i)T^{2}
47 1+0.231iT−47T2 1 + 0.231iT - 47T^{2}
53 1+5.16iT−53T2 1 + 5.16iT - 53T^{2}
59 1+(1.57−2.73i)T+(−29.5−51.0i)T2 1 + (1.57 - 2.73i)T + (-29.5 - 51.0i)T^{2}
61 1+(−7.73−4.46i)T+(30.5+52.8i)T2 1 + (-7.73 - 4.46i)T + (30.5 + 52.8i)T^{2}
67 1+(4.31+7.46i)T+(−33.5+58.0i)T2 1 + (4.31 + 7.46i)T + (-33.5 + 58.0i)T^{2}
71 1+(13.9+8.05i)T+(35.5+61.4i)T2 1 + (13.9 + 8.05i)T + (35.5 + 61.4i)T^{2}
73 1−10.1iT−73T2 1 - 10.1iT - 73T^{2}
79 1−6.72T+79T2 1 - 6.72T + 79T^{2}
83 1−5.31T+83T2 1 - 5.31T + 83T^{2}
89 1+(8.75−5.05i)T+(44.5−77.0i)T2 1 + (8.75 - 5.05i)T + (44.5 - 77.0i)T^{2}
97 1+(−7.23−4.17i)T+(48.5+84.0i)T2 1 + (-7.23 - 4.17i)T + (48.5 + 84.0i)T^{2}
show more
show less
   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

−10.61053363384611251462164927713, −10.14391725509547188687931430295, −9.438424660600124487287438869534, −8.428400044393655265662666919493, −7.42128913618075521812661889681, −6.42832018732874668654135931764, −5.05859554439950563673135243455, −3.91078642967024155841616901044, −2.85093360255977855582699811153, −1.43739499758107553542587657083, 0.70482798845407586880106934171, 2.87086380099769589696957779339, 4.23478062363591679304462358788, 5.77316163353839840285157021592, 6.33380313077150501699720047014, 6.78105655766776464751469306996, 8.508426688111045688884127249452, 9.043382240712240118277642028403, 9.442793546171058397160939944775, 10.75229998936710925501384024730

Graph of the ZZ-function along the critical line