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Properties

Label 2-650-5.4-c5-0-79
Degree 22
Conductor 650650
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 104.249104.249
Root an. cond. 10.210210.2102
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 3.22i·3-s − 16·4-s + 12.8·6-s − 200. i·7-s − 64i·8-s + 232.·9-s − 530.·11-s + 51.5i·12-s + 169i·13-s + 803.·14-s + 256·16-s − 15.1i·17-s + 930. i·18-s + 392.·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.206i·3-s − 0.5·4-s + 0.146·6-s − 1.54i·7-s − 0.353i·8-s + 0.957·9-s − 1.32·11-s + 0.103i·12-s + 0.277i·13-s + 1.09·14-s + 0.250·16-s − 0.0126i·17-s + 0.676i·18-s + 0.249·19-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+5/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 104.249104.249
Root analytic conductor: 10.210210.2102
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ650(599,⋅)\chi_{650} (599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :5/2), −0.447+0.894i)(2,\ 650,\ (\ :5/2),\ -0.447 + 0.894i)

Particular Values

L(3)L(3) ≈\approx 1.1756399591.175639959
L(12)L(\frac12) ≈\approx 1.1756399591.175639959
L(72)L(\frac{7}{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 1−4iT 1 - 4iT
5 1 1
13 1−169iT 1 - 169iT
good3 1+3.22iT−243T2 1 + 3.22iT - 243T^{2}
7 1+200.iT−1.68e4T2 1 + 200. iT - 1.68e4T^{2}
11 1+530.T+1.61e5T2 1 + 530.T + 1.61e5T^{2}
17 1+15.1iT−1.41e6T2 1 + 15.1iT - 1.41e6T^{2}
19 1−392.T+2.47e6T2 1 - 392.T + 2.47e6T^{2}
23 1+2.63e3iT−6.43e6T2 1 + 2.63e3iT - 6.43e6T^{2}
29 1−7.13e3T+2.05e7T2 1 - 7.13e3T + 2.05e7T^{2}
31 1−6.82e3T+2.86e7T2 1 - 6.82e3T + 2.86e7T^{2}
37 1+1.32e4iT−6.93e7T2 1 + 1.32e4iT - 6.93e7T^{2}
41 1−3.21e3T+1.15e8T2 1 - 3.21e3T + 1.15e8T^{2}
43 1−1.10e4iT−1.47e8T2 1 - 1.10e4iT - 1.47e8T^{2}
47 1−9.25e3iT−2.29e8T2 1 - 9.25e3iT - 2.29e8T^{2}
53 1+3.52e3iT−4.18e8T2 1 + 3.52e3iT - 4.18e8T^{2}
59 1+4.03e4T+7.14e8T2 1 + 4.03e4T + 7.14e8T^{2}
61 1+4.42e4T+8.44e8T2 1 + 4.42e4T + 8.44e8T^{2}
67 1−7.07e3iT−1.35e9T2 1 - 7.07e3iT - 1.35e9T^{2}
71 1+3.62e4T+1.80e9T2 1 + 3.62e4T + 1.80e9T^{2}
73 1−4.10e4iT−2.07e9T2 1 - 4.10e4iT - 2.07e9T^{2}
79 1−1.94e4T+3.07e9T2 1 - 1.94e4T + 3.07e9T^{2}
83 1+6.75e4iT−3.93e9T2 1 + 6.75e4iT - 3.93e9T^{2}
89 1+3.33e4T+5.58e9T2 1 + 3.33e4T + 5.58e9T^{2}
97 1+2.20e3iT−8.58e9T2 1 + 2.20e3iT - 8.58e9T^{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

−9.597316264009758334032716786219, −8.277769652208717518294035809061, −7.61498839742408040073153062504, −6.98757781218594973133115870854, −6.13172444207821644923400486541, −4.68940879937535060625406298549, −4.30542046913951442647374910370, −2.85084614691639572370735438477, −1.22022309103391759501140567243, −0.27551089067402655469312379277, 1.28978000147736598621030073539, 2.47354201959764268132148851212, 3.17746977429690549705256963460, 4.66197663612106823366654314072, 5.27502655931792704818662700044, 6.32225895189913788447326419851, 7.71027238727486442320775011105, 8.440191531470252838850882259978, 9.382865357404957939439336053971, 10.10941686365408262597627426219

Graph of the ZZ-function along the critical line