↑

Properties

Label 2-520-104.101-c1-0-26
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(101,⋅)\chi_{520} (101, \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

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

Graph of the ZZ-function along the critical line