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Properties

Label 2-5e2-5.2-c2-0-1
Degree 22
Conductor 2525
Sign 0.437+0.899i0.437 + 0.899i
Analytic cond. 0.6812000.681200
Root an. cond. 0.8253480.825348
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)2-s + (1.22 − 1.22i)3-s − 1.00i·4-s − 2.99·6-s + (4.89 + 4.89i)7-s + (−6.12 + 6.12i)8-s + 6i·9-s − 3·11-s + (−1.22 − 1.22i)12-s + (7.34 − 7.34i)13-s − 11.9i·14-s + 10.9·16-s + (−13.4 − 13.4i)17-s + (7.34 − 7.34i)18-s + 5i·19-s + ⋯
L(s)  = 1  + (−0.612 − 0.612i)2-s + (0.408 − 0.408i)3-s − 0.250i·4-s − 0.499·6-s + (0.699 + 0.699i)7-s + (−0.765 + 0.765i)8-s + 0.666i·9-s − 0.272·11-s + (−0.102 − 0.102i)12-s + (0.565 − 0.565i)13-s − 0.857i·14-s + 0.687·16-s + (−0.792 − 0.792i)17-s + (0.408 − 0.408i)18-s + 0.263i·19-s + ⋯

Functional equation

Λ(s)=(25s/2 ΓC(s) L(s)=((0.437+0.899i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(25s/2 ΓC(s+1) L(s)=((0.437+0.899i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2525    =    525^{2}
Sign: 0.437+0.899i0.437 + 0.899i
Analytic conductor: 0.6812000.681200
Root analytic conductor: 0.8253480.825348
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ25(7,⋅)\chi_{25} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 25, ( :1), 0.437+0.899i)(2,\ 25,\ (\ :1),\ 0.437 + 0.899i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.672853−0.420861i0.672853 - 0.420861i
L(12)L(\frac12) ≈\approx 0.672853−0.420861i0.672853 - 0.420861i
L(2)L(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)
bad5 1 1
good2 1+(1.22+1.22i)T+4iT2 1 + (1.22 + 1.22i)T + 4iT^{2}
3 1+(−1.22+1.22i)T−9iT2 1 + (-1.22 + 1.22i)T - 9iT^{2}
7 1+(−4.89−4.89i)T+49iT2 1 + (-4.89 - 4.89i)T + 49iT^{2}
11 1+3T+121T2 1 + 3T + 121T^{2}
13 1+(−7.34+7.34i)T−169iT2 1 + (-7.34 + 7.34i)T - 169iT^{2}
17 1+(13.4+13.4i)T+289iT2 1 + (13.4 + 13.4i)T + 289iT^{2}
19 1−5iT−361T2 1 - 5iT - 361T^{2}
23 1+(17.1−17.1i)T−529iT2 1 + (17.1 - 17.1i)T - 529iT^{2}
29 1+30iT−841T2 1 + 30iT - 841T^{2}
31 1+38T+961T2 1 + 38T + 961T^{2}
37 1+(19.5+19.5i)T+1.36e3iT2 1 + (19.5 + 19.5i)T + 1.36e3iT^{2}
41 1−57T+1.68e3T2 1 - 57T + 1.68e3T^{2}
43 1+(4.89−4.89i)T−1.84e3iT2 1 + (4.89 - 4.89i)T - 1.84e3iT^{2}
47 1+(7.34+7.34i)T+2.20e3iT2 1 + (7.34 + 7.34i)T + 2.20e3iT^{2}
53 1+(−31.8+31.8i)T−2.80e3iT2 1 + (-31.8 + 31.8i)T - 2.80e3iT^{2}
59 1−90iT−3.48e3T2 1 - 90iT - 3.48e3T^{2}
61 1+28T+3.72e3T2 1 + 28T + 3.72e3T^{2}
67 1+(−47.7−47.7i)T+4.48e3iT2 1 + (-47.7 - 47.7i)T + 4.48e3iT^{2}
71 1−42T+5.04e3T2 1 - 42T + 5.04e3T^{2}
73 1+(−13.4+13.4i)T−5.32e3iT2 1 + (-13.4 + 13.4i)T - 5.32e3iT^{2}
79 1+80iT−6.24e3T2 1 + 80iT - 6.24e3T^{2}
83 1+(−111.+111.i)T−6.88e3iT2 1 + (-111. + 111. i)T - 6.88e3iT^{2}
89 1+15iT−7.92e3T2 1 + 15iT - 7.92e3T^{2}
97 1+(−53.8−53.8i)T+9.40e3iT2 1 + (-53.8 - 53.8i)T + 9.40e3iT^{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

−17.80124358464963284926715165367, −15.87327121129670108706619144263, −14.61987166921431862955652160781, −13.38291394650940071904926706610, −11.72130267965928507532906959660, −10.65256669380755182035270122951, −9.075493128417653782841684094869, −7.902577928817560403427339720800, −5.50779105914296339937998810838, −2.18276126975121981967295505990, 3.96617731422108493669297013406, 6.66659017825642368793563998673, 8.181050810439558547923718676808, 9.231979411563966906372676961873, 10.87036483367697919382493644174, 12.57910233808340025768232932523, 14.17692961093489109108404805037, 15.35474335332985474713232506729, 16.42661081484645235263474578445, 17.56119575775530289336883258965

Graph of the ZZ-function along the critical line