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Properties

Label 2-520-104.101-c1-0-15
Degree 22
Conductor 520520
Sign −0.759−0.650i-0.759 - 0.650i
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  + (−0.683 + 1.23i)2-s + (−2.80 + 1.62i)3-s + (−1.06 − 1.69i)4-s − 5-s + (−0.0886 − 4.58i)6-s + (3.76 + 2.17i)7-s + (2.82 − 0.163i)8-s + (3.75 − 6.50i)9-s + (0.683 − 1.23i)10-s + (1.73 + 3.01i)11-s + (5.73 + 3.02i)12-s + (3.51 + 0.821i)13-s + (−5.26 + 3.17i)14-s + (2.80 − 1.62i)15-s + (−1.72 + 3.60i)16-s + (−0.0871 + 0.150i)17-s + ⋯
L(s)  = 1  + (−0.483 + 0.875i)2-s + (−1.62 + 0.936i)3-s + (−0.533 − 0.846i)4-s − 0.447·5-s + (−0.0361 − 1.87i)6-s + (1.42 + 0.822i)7-s + (0.998 − 0.0579i)8-s + (1.25 − 2.16i)9-s + (0.216 − 0.391i)10-s + (0.524 + 0.908i)11-s + (1.65 + 0.872i)12-s + (0.973 + 0.227i)13-s + (−1.40 + 0.849i)14-s + (0.725 − 0.418i)15-s + (−0.431 + 0.902i)16-s + (−0.0211 + 0.0366i)17-s + ⋯

Functional equation

Λ(s)=(520s/2 ΓC(s) L(s)=((−0.759−0.650i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(520s/2 ΓC(s+1/2) L(s)=((−0.759−0.650i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 520520    =    23⋅5⋅132^{3} \cdot 5 \cdot 13
Sign: −0.759−0.650i-0.759 - 0.650i
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), −0.759−0.650i)(2,\ 520,\ (\ :1/2),\ -0.759 - 0.650i)

Particular Values

L(1)L(1) ≈\approx 0.250899+0.678562i0.250899 + 0.678562i
L(12)L(\frac12) ≈\approx 0.250899+0.678562i0.250899 + 0.678562i
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+(0.683−1.23i)T 1 + (0.683 - 1.23i)T
5 1+T 1 + T
13 1+(−3.51−0.821i)T 1 + (-3.51 - 0.821i)T
good3 1+(2.80−1.62i)T+(1.5−2.59i)T2 1 + (2.80 - 1.62i)T + (1.5 - 2.59i)T^{2}
7 1+(−3.76−2.17i)T+(3.5+6.06i)T2 1 + (-3.76 - 2.17i)T + (3.5 + 6.06i)T^{2}
11 1+(−1.73−3.01i)T+(−5.5+9.52i)T2 1 + (-1.73 - 3.01i)T + (-5.5 + 9.52i)T^{2}
17 1+(0.0871−0.150i)T+(−8.5−14.7i)T2 1 + (0.0871 - 0.150i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.52+6.10i)T+(−9.5−16.4i)T2 1 + (-3.52 + 6.10i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.971−1.68i)T+(−11.5+19.9i)T2 1 + (-0.971 - 1.68i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.50+1.44i)T+(14.5−25.1i)T2 1 + (-2.50 + 1.44i)T + (14.5 - 25.1i)T^{2}
31 1−4.00iT−31T2 1 - 4.00iT - 31T^{2}
37 1+(3.07+5.33i)T+(−18.5+32.0i)T2 1 + (3.07 + 5.33i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.76−2.17i)T+(20.5−35.5i)T2 1 + (3.76 - 2.17i)T + (20.5 - 35.5i)T^{2}
43 1+(−7.49−4.32i)T+(21.5+37.2i)T2 1 + (-7.49 - 4.32i)T + (21.5 + 37.2i)T^{2}
47 1+3.80iT−47T2 1 + 3.80iT - 47T^{2}
53 1−6.77iT−53T2 1 - 6.77iT - 53T^{2}
59 1+(0.345−0.598i)T+(−29.5−51.0i)T2 1 + (0.345 - 0.598i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.53+2.04i)T+(30.5+52.8i)T2 1 + (3.53 + 2.04i)T + (30.5 + 52.8i)T^{2}
67 1+(−3.84−6.65i)T+(−33.5+58.0i)T2 1 + (-3.84 - 6.65i)T + (-33.5 + 58.0i)T^{2}
71 1+(−5.70−3.29i)T+(35.5+61.4i)T2 1 + (-5.70 - 3.29i)T + (35.5 + 61.4i)T^{2}
73 1+5.78iT−73T2 1 + 5.78iT - 73T^{2}
79 1+4.84T+79T2 1 + 4.84T + 79T^{2}
83 1+10.7T+83T2 1 + 10.7T + 83T^{2}
89 1+(7.02−4.05i)T+(44.5−77.0i)T2 1 + (7.02 - 4.05i)T + (44.5 - 77.0i)T^{2}
97 1+(−4.51−2.60i)T+(48.5+84.0i)T2 1 + (-4.51 - 2.60i)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

−11.27172413891996994829742709864, −10.41322322623461078112313540408, −9.330232856766680740065864753232, −8.742989799306552269371930419989, −7.42934935952395695739629937996, −6.55284887351689353780499205280, −5.52844139116810097895252530323, −4.88896513982967880740899364881, −4.19378364810605714050278487839, −1.23041464378447211404285504359, 0.843136012976828336379892319886, 1.55413194417071643600485904752, 3.74975773846348963265573064791, 4.79927490862892479415182460376, 5.84437711928427085088532813799, 7.06190511452220821569090037581, 7.896595382970192789991792768019, 8.462472877835546868351544281465, 10.20189549406142803925910604144, 10.89403145270831062294725999547

Graph of the ZZ-function along the critical line