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Properties

Label 2-520-104.101-c1-0-10
Degree 22
Conductor 520520
Sign 0.917−0.397i0.917 - 0.397i
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  + (−1.24 − 0.669i)2-s + (−2.59 + 1.49i)3-s + (1.10 + 1.66i)4-s − 5-s + (4.23 − 0.128i)6-s + (−0.668 − 0.385i)7-s + (−0.257 − 2.81i)8-s + (2.98 − 5.16i)9-s + (1.24 + 0.669i)10-s + (0.183 + 0.317i)11-s + (−5.35 − 2.67i)12-s + (−2.71 − 2.36i)13-s + (0.574 + 0.928i)14-s + (2.59 − 1.49i)15-s + (−1.56 + 3.68i)16-s + (1.37 − 2.37i)17-s + ⋯
L(s)  = 1  + (−0.880 − 0.473i)2-s + (−1.49 + 0.864i)3-s + (0.551 + 0.834i)4-s − 0.447·5-s + (1.72 − 0.0525i)6-s + (−0.252 − 0.145i)7-s + (−0.0911 − 0.995i)8-s + (0.994 − 1.72i)9-s + (0.393 + 0.211i)10-s + (0.0552 + 0.0956i)11-s + (−1.54 − 0.771i)12-s + (−0.753 − 0.657i)13-s + (0.153 + 0.248i)14-s + (0.669 − 0.386i)15-s + (−0.391 + 0.920i)16-s + (0.332 − 0.575i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((0.917−0.397i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((0.917−0.397i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 0.917−0.397i0.917 - 0.397i
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), 0.917−0.397i)(2,\ 520,\ (\ :1/2),\ 0.917 - 0.397i)

Particular Values

L(1)L(1) ≈\approx 0.399823+0.0828691i0.399823 + 0.0828691i
L(12)L(\frac12) ≈\approx 0.399823+0.0828691i0.399823 + 0.0828691i
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+(1.24+0.669i)T 1 + (1.24 + 0.669i)T
5 1+T 1 + T
13 1+(2.71+2.36i)T 1 + (2.71 + 2.36i)T
good3 1+(2.59−1.49i)T+(1.5−2.59i)T2 1 + (2.59 - 1.49i)T + (1.5 - 2.59i)T^{2}
7 1+(0.668+0.385i)T+(3.5+6.06i)T2 1 + (0.668 + 0.385i)T + (3.5 + 6.06i)T^{2}
11 1+(−0.183−0.317i)T+(−5.5+9.52i)T2 1 + (-0.183 - 0.317i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1.37+2.37i)T+(−8.5−14.7i)T2 1 + (-1.37 + 2.37i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.199−0.345i)T+(−9.5−16.4i)T2 1 + (0.199 - 0.345i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.390+0.676i)T+(−11.5+19.9i)T2 1 + (0.390 + 0.676i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.82+1.05i)T+(14.5−25.1i)T2 1 + (-1.82 + 1.05i)T + (14.5 - 25.1i)T^{2}
31 1−9.83iT−31T2 1 - 9.83iT - 31T^{2}
37 1+(−4.55−7.88i)T+(−18.5+32.0i)T2 1 + (-4.55 - 7.88i)T + (-18.5 + 32.0i)T^{2}
41 1+(−6.57+3.79i)T+(20.5−35.5i)T2 1 + (-6.57 + 3.79i)T + (20.5 - 35.5i)T^{2}
43 1+(−5.38−3.10i)T+(21.5+37.2i)T2 1 + (-5.38 - 3.10i)T + (21.5 + 37.2i)T^{2}
47 1−8.57iT−47T2 1 - 8.57iT - 47T^{2}
53 1+4.85iT−53T2 1 + 4.85iT - 53T^{2}
59 1+(−6.27+10.8i)T+(−29.5−51.0i)T2 1 + (-6.27 + 10.8i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.23−0.712i)T+(30.5+52.8i)T2 1 + (-1.23 - 0.712i)T + (30.5 + 52.8i)T^{2}
67 1+(−2.10−3.63i)T+(−33.5+58.0i)T2 1 + (-2.10 - 3.63i)T + (-33.5 + 58.0i)T^{2}
71 1+(−1.22−0.707i)T+(35.5+61.4i)T2 1 + (-1.22 - 0.707i)T + (35.5 + 61.4i)T^{2}
73 1−16.3iT−73T2 1 - 16.3iT - 73T^{2}
79 1+4.89T+79T2 1 + 4.89T + 79T^{2}
83 1−2.70T+83T2 1 - 2.70T + 83T^{2}
89 1+(10.0−5.82i)T+(44.5−77.0i)T2 1 + (10.0 - 5.82i)T + (44.5 - 77.0i)T^{2}
97 1+(−11.2−6.48i)T+(48.5+84.0i)T2 1 + (-11.2 - 6.48i)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

−10.92965539200928051049301081009, −10.02015324749927022610244460372, −9.687749270966646275447590009993, −8.391046963813284686333732205388, −7.27803871036506343188789810422, −6.44159044929553179640209603234, −5.22339292356978175674787377440, −4.24187551814913355895296024280, −3.02832494781368753241627728914, −0.75287887081118375401535926109, 0.64134454576838714749813726514, 2.15262117236275886992444410208, 4.48862053096954855986474601595, 5.71999483110753094163265802037, 6.23982683455060525040749905597, 7.28978094072647894073211456724, 7.71352832253405875316080711042, 9.024609824169323005593419231669, 10.02051018443913870652362499786, 10.93231433691617764792891963728

Graph of the ZZ-function along the critical line