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Properties

Label 2-520-104.69-c1-0-24
Degree 22
Conductor 520520
Sign 0.974+0.224i0.974 + 0.224i
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  + (−0.0430 − 1.41i)2-s + (2.59 + 1.49i)3-s + (−1.99 + 0.121i)4-s + 5-s + (2.00 − 3.73i)6-s + (−0.668 + 0.385i)7-s + (0.257 + 2.81i)8-s + (2.98 + 5.16i)9-s + (−0.0430 − 1.41i)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 + (3.97 − 0.485i)16-s + (1.37 + 2.37i)17-s + ⋯
L(s)  = 1  + (−0.0304 − 0.999i)2-s + (1.49 + 0.864i)3-s + (−0.998 + 0.0607i)4-s + 0.447·5-s + (0.818 − 1.52i)6-s + (−0.252 + 0.145i)7-s + (0.0911 + 0.995i)8-s + (0.994 + 1.72i)9-s + (−0.0135 − 0.447i)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 + (0.992 − 0.121i)16-s + (0.332 + 0.575i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((0.974+0.224i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((0.974+0.224i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: 0.974+0.224i0.974 + 0.224i
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), 0.974+0.224i)(2,\ 520,\ (\ :1/2),\ 0.974 + 0.224i)

Particular Values

L(1)L(1) ≈\approx 2.19933−0.249899i2.19933 - 0.249899i
L(12)L(\frac12) ≈\approx 2.19933−0.249899i2.19933 - 0.249899i
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+(0.0430+1.41i)T 1 + (0.0430 + 1.41i)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

−10.52645306273854197021425214715, −9.878560308416904263005613558514, −9.334912849098707373355013030762, −8.406805295495744450394235533397, −7.86370717026659879194503477720, −5.95518785724294527360351597503, −4.72604376305583212123962736152, −3.68004631478244396131897317531, −2.98147159495619936159925052319, −1.82115802237513887872885982519, 1.42791936693660863274840154192, 3.00628778133818768570685640997, 4.06074010338716614691191614362, 5.56585829426747353214057327105, 6.72682475722575212115911754995, 7.21853659583952590755064474213, 8.210309570014313917684543542686, 8.966431446236856839968579909888, 9.425750353122368022044346281267, 10.60281271894754530288946509840

Graph of the ZZ-function along the critical line