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Properties

Label 2-260-65.49-c1-0-3
Degree 22
Conductor 260260
Sign 0.956−0.292i0.956 - 0.292i
Analytic cond. 2.076112.07611
Root an. cond. 1.440871.44087
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  + (−0.180 + 0.104i)3-s + (1.60 + 1.56i)5-s + (1.72 − 2.99i)7-s + (−1.47 + 2.56i)9-s + (0.625 − 0.360i)11-s + (3.18 + 1.69i)13-s + (−0.452 − 0.115i)15-s + (−3.10 − 1.79i)17-s + (6.51 + 3.76i)19-s + 0.721i·21-s + (2.05 − 1.18i)23-s + (0.125 + 4.99i)25-s − 1.24i·27-s + (−3.68 − 6.37i)29-s + 0.668i·31-s + ⋯
L(s)  = 1  + (−0.104 + 0.0602i)3-s + (0.715 + 0.698i)5-s + (0.653 − 1.13i)7-s + (−0.492 + 0.853i)9-s + (0.188 − 0.108i)11-s + (0.883 + 0.468i)13-s + (−0.116 − 0.0297i)15-s + (−0.754 − 0.435i)17-s + (1.49 + 0.862i)19-s + 0.157i·21-s + (0.428 − 0.247i)23-s + (0.0250 + 0.999i)25-s − 0.239i·27-s + (−0.683 − 1.18i)29-s + 0.120i·31-s + ⋯

Functional equation

Λ(s)=(260s/2 ΓC(s) L(s)=((0.956−0.292i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(260s/2 ΓC(s+1/2) L(s)=((0.956−0.292i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 260260    =    22⋅5⋅132^{2} \cdot 5 \cdot 13
Sign: 0.956−0.292i0.956 - 0.292i
Analytic conductor: 2.076112.07611
Root analytic conductor: 1.440871.44087
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ260(49,⋅)\chi_{260} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 260, ( :1/2), 0.956−0.292i)(2,\ 260,\ (\ :1/2),\ 0.956 - 0.292i)

Particular Values

L(1)L(1) ≈\approx 1.41441+0.211621i1.41441 + 0.211621i
L(12)L(\frac12) ≈\approx 1.41441+0.211621i1.41441 + 0.211621i
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
5 1+(−1.60−1.56i)T 1 + (-1.60 - 1.56i)T
13 1+(−3.18−1.69i)T 1 + (-3.18 - 1.69i)T
good3 1+(0.180−0.104i)T+(1.5−2.59i)T2 1 + (0.180 - 0.104i)T + (1.5 - 2.59i)T^{2}
7 1+(−1.72+2.99i)T+(−3.5−6.06i)T2 1 + (-1.72 + 2.99i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.625+0.360i)T+(5.5−9.52i)T2 1 + (-0.625 + 0.360i)T + (5.5 - 9.52i)T^{2}
17 1+(3.10+1.79i)T+(8.5+14.7i)T2 1 + (3.10 + 1.79i)T + (8.5 + 14.7i)T^{2}
19 1+(−6.51−3.76i)T+(9.5+16.4i)T2 1 + (-6.51 - 3.76i)T + (9.5 + 16.4i)T^{2}
23 1+(−2.05+1.18i)T+(11.5−19.9i)T2 1 + (-2.05 + 1.18i)T + (11.5 - 19.9i)T^{2}
29 1+(3.68+6.37i)T+(−14.5+25.1i)T2 1 + (3.68 + 6.37i)T + (-14.5 + 25.1i)T^{2}
31 1−0.668iT−31T2 1 - 0.668iT - 31T^{2}
37 1+(3.36+5.82i)T+(−18.5+32.0i)T2 1 + (3.36 + 5.82i)T + (-18.5 + 32.0i)T^{2}
41 1+(6.32−3.65i)T+(20.5−35.5i)T2 1 + (6.32 - 3.65i)T + (20.5 - 35.5i)T^{2}
43 1+(7.06+4.07i)T+(21.5+37.2i)T2 1 + (7.06 + 4.07i)T + (21.5 + 37.2i)T^{2}
47 1+6.40T+47T2 1 + 6.40T + 47T^{2}
53 1−11.7iT−53T2 1 - 11.7iT - 53T^{2}
59 1+(4.20+2.42i)T+(29.5+51.0i)T2 1 + (4.20 + 2.42i)T + (29.5 + 51.0i)T^{2}
61 1+(−2.68+4.64i)T+(−30.5−52.8i)T2 1 + (-2.68 + 4.64i)T + (-30.5 - 52.8i)T^{2}
67 1+(7.80+13.5i)T+(−33.5+58.0i)T2 1 + (7.80 + 13.5i)T + (-33.5 + 58.0i)T^{2}
71 1+(4.10+2.37i)T+(35.5+61.4i)T2 1 + (4.10 + 2.37i)T + (35.5 + 61.4i)T^{2}
73 1−6.36T+73T2 1 - 6.36T + 73T^{2}
79 1−5.20T+79T2 1 - 5.20T + 79T^{2}
83 1+13.3T+83T2 1 + 13.3T + 83T^{2}
89 1+(−4.20+2.42i)T+(44.5−77.0i)T2 1 + (-4.20 + 2.42i)T + (44.5 - 77.0i)T^{2}
97 1+(1.83−3.17i)T+(−48.5−84.0i)T2 1 + (1.83 - 3.17i)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

−11.66333559236957199128557835438, −11.03196722298165007157851563574, −10.34069826486997929320441687886, −9.273710149678043273473434517198, −7.999217340061549933878430745229, −7.11646174144086377231267634257, −5.99424160715074085990629934944, −4.84011212626120241442203237095, −3.43870905634417617010528586958, −1.75683604869581378162560476597, 1.51819065312308178021858497700, 3.19164129382976587013653879256, 5.02328468854763650252714725260, 5.68635478371143725221821028120, 6.78859440677974890382921001345, 8.519502785247397476504067147677, 8.847702490945227986803086233370, 9.848422234464662102394945589102, 11.28673979965643109570922009825, 11.84051520525299707894034333150

Graph of the ZZ-function along the critical line