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Properties

Label 2-35e2-7.5-c0-0-0
Degree 22
Conductor 12251225
Sign −0.895−0.444i-0.895 - 0.444i
Analytic cond. 0.6113540.611354
Root an. cond. 0.7818910.781891
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (−0.499 − 0.866i)18-s − 0.999·22-s + (−0.5 + 0.866i)23-s − 29-s + (−0.5 + 0.866i)37-s + 43-s + (−0.499 − 0.866i)46-s + (1 + 1.73i)53-s + (0.5 − 0.866i)58-s + 64-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)16-s + (−0.499 − 0.866i)18-s − 0.999·22-s + (−0.5 + 0.866i)23-s − 29-s + (−0.5 + 0.866i)37-s + 43-s + (−0.499 − 0.866i)46-s + (1 + 1.73i)53-s + (0.5 − 0.866i)58-s + 64-s + ⋯

Functional equation

Λ(s)=(1225s/2 ΓC(s) L(s)=((−0.895−0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1225s/2 ΓC(s) L(s)=((−0.895−0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12251225    =    52⋅725^{2} \cdot 7^{2}
Sign: −0.895−0.444i-0.895 - 0.444i
Analytic conductor: 0.6113540.611354
Root analytic conductor: 0.7818910.781891
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1225(901,⋅)\chi_{1225} (901, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1225, ( :0), −0.895−0.444i)(2,\ 1225,\ (\ :0),\ -0.895 - 0.444i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.69695932400.6969593240
L(12)L(\frac12) ≈\approx 0.69695932400.6969593240
L(1)L(1) 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)
bad5 1 1
7 1 1
good2 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
3 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
29 1+T+T2 1 + T + T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−T+T2 1 - T + T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1+T+T2 1 + T + T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1−T2 1 - 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

−10.03313300817045812074929488173, −9.229758580470922955291942117830, −8.576136475883728372990583116844, −7.59842176902218999053938089307, −7.30725607402941585738144896994, −6.19099761555383735231315200215, −5.49525113082812853994718989058, −4.37372423176520755756869262364, −3.15985079605847204754569431114, −1.95256991266311545215048999760, 0.69949591763161655647229985242, 2.10925468469655311819240858735, 3.20266447089468753644929897930, 3.98645908157874496953427827379, 5.57428619416559246494646842947, 6.13763063646913018186982607990, 7.04006346212564303124405887976, 8.372531608678366215526791706188, 8.924255916202679686666584223595, 9.578040823003165920550467263826

Graph of the ZZ-function along the critical line