↑

Properties

Label 2-520-104.69-c1-0-38
Degree 22
Conductor 520520
Sign 0.236+0.971i0.236 + 0.971i
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  + (0.164 − 1.40i)2-s + (1.79 + 1.03i)3-s + (−1.94 − 0.462i)4-s − 5-s + (1.75 − 2.35i)6-s + (3.65 − 2.10i)7-s + (−0.969 + 2.65i)8-s + (0.658 + 1.14i)9-s + (−0.164 + 1.40i)10-s + (2.34 − 4.05i)11-s + (−3.02 − 2.85i)12-s + (−3.60 + 0.189i)13-s + (−2.36 − 5.48i)14-s + (−1.79 − 1.03i)15-s + (3.57 + 1.79i)16-s + (1.44 + 2.49i)17-s + ⋯
L(s)  = 1  + (0.116 − 0.993i)2-s + (1.03 + 0.599i)3-s + (−0.972 − 0.231i)4-s − 0.447·5-s + (0.716 − 0.961i)6-s + (1.38 − 0.797i)7-s + (−0.342 + 0.939i)8-s + (0.219 + 0.380i)9-s + (−0.0520 + 0.444i)10-s + (0.705 − 1.22i)11-s + (−0.872 − 0.823i)12-s + (−0.998 + 0.0524i)13-s + (−0.631 − 1.46i)14-s + (−0.464 − 0.268i)15-s + (0.893 + 0.449i)16-s + (0.349 + 0.605i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((0.236+0.971i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((0.236+0.971i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 0.236+0.971i0.236 + 0.971i
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), 0.236+0.971i)(2,\ 520,\ (\ :1/2),\ 0.236 + 0.971i)

Particular Values

L(1)L(1) ≈\approx 1.58069−1.24169i1.58069 - 1.24169i
L(12)L(\frac12) ≈\approx 1.58069−1.24169i1.58069 - 1.24169i
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+(−0.164+1.40i)T 1 + (-0.164 + 1.40i)T
5 1+T 1 + T
13 1+(3.60−0.189i)T 1 + (3.60 - 0.189i)T
good3 1+(−1.79−1.03i)T+(1.5+2.59i)T2 1 + (-1.79 - 1.03i)T + (1.5 + 2.59i)T^{2}
7 1+(−3.65+2.10i)T+(3.5−6.06i)T2 1 + (-3.65 + 2.10i)T + (3.5 - 6.06i)T^{2}
11 1+(−2.34+4.05i)T+(−5.5−9.52i)T2 1 + (-2.34 + 4.05i)T + (-5.5 - 9.52i)T^{2}
17 1+(−1.44−2.49i)T+(−8.5+14.7i)T2 1 + (-1.44 - 2.49i)T + (-8.5 + 14.7i)T^{2}
19 1+(−3.09−5.36i)T+(−9.5+16.4i)T2 1 + (-3.09 - 5.36i)T + (-9.5 + 16.4i)T^{2}
23 1+(−3.36+5.82i)T+(−11.5−19.9i)T2 1 + (-3.36 + 5.82i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.64+2.68i)T+(14.5+25.1i)T2 1 + (4.64 + 2.68i)T + (14.5 + 25.1i)T^{2}
31 1−0.540iT−31T2 1 - 0.540iT - 31T^{2}
37 1+(−0.815+1.41i)T+(−18.5−32.0i)T2 1 + (-0.815 + 1.41i)T + (-18.5 - 32.0i)T^{2}
41 1+(−1.36−0.789i)T+(20.5+35.5i)T2 1 + (-1.36 - 0.789i)T + (20.5 + 35.5i)T^{2}
43 1+(4.99−2.88i)T+(21.5−37.2i)T2 1 + (4.99 - 2.88i)T + (21.5 - 37.2i)T^{2}
47 1+4.24iT−47T2 1 + 4.24iT - 47T^{2}
53 1−10.7iT−53T2 1 - 10.7iT - 53T^{2}
59 1+(−6.90−11.9i)T+(−29.5+51.0i)T2 1 + (-6.90 - 11.9i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.45−3.72i)T+(30.5−52.8i)T2 1 + (6.45 - 3.72i)T + (30.5 - 52.8i)T^{2}
67 1+(5.52−9.57i)T+(−33.5−58.0i)T2 1 + (5.52 - 9.57i)T + (-33.5 - 58.0i)T^{2}
71 1+(8.93−5.15i)T+(35.5−61.4i)T2 1 + (8.93 - 5.15i)T + (35.5 - 61.4i)T^{2}
73 1−8.13iT−73T2 1 - 8.13iT - 73T^{2}
79 1−1.61T+79T2 1 - 1.61T + 79T^{2}
83 1+3.34T+83T2 1 + 3.34T + 83T^{2}
89 1+(2.76+1.59i)T+(44.5+77.0i)T2 1 + (2.76 + 1.59i)T + (44.5 + 77.0i)T^{2}
97 1+(12.2−7.09i)T+(48.5−84.0i)T2 1 + (12.2 - 7.09i)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

−10.63915080192071895087921246010, −9.963068225141115831296842798094, −8.878885258517545813702322271973, −8.325656035545938346213757016640, −7.53835656986924000920804129674, −5.66834811119193258009147688515, −4.39284365101256594036722447381, −3.87137323456400241489416727013, −2.81105654334428395697052199666, −1.24682084709249682841399040398, 1.79368889650067509760944600728, 3.19450128928234869027592457659, 4.76626397580538026480148602963, 5.21862668094379599661123532025, 7.03685339576331675369178819396, 7.43653995082466444645552968475, 8.125426306612772650557729042919, 9.128312819851898963906718324157, 9.520449079200257559271561267291, 11.34812029996598968280292939024

Graph of the ZZ-function along the critical line