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Properties

Label 2-55-55.13-c1-0-3
Degree 22
Conductor 5555
Sign 0.875+0.482i0.875 + 0.482i
Analytic cond. 0.4391770.439177
Root an. cond. 0.6627040.662704
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  + (1.30 − 0.665i)2-s + (−0.822 − 0.130i)3-s + (0.0875 − 0.120i)4-s + (−0.233 − 2.22i)5-s + (−1.16 + 0.377i)6-s + (0.659 + 4.16i)7-s + (−0.424 + 2.68i)8-s + (−2.19 − 0.712i)9-s + (−1.78 − 2.74i)10-s + (−0.920 − 3.18i)11-s + (−0.0877 + 0.0877i)12-s + (−0.420 − 0.824i)13-s + (3.63 + 4.99i)14-s + (−0.0979 + 1.85i)15-s + (1.32 + 4.06i)16-s + (0.875 − 1.71i)17-s + ⋯
L(s)  = 1  + (0.923 − 0.470i)2-s + (−0.474 − 0.0751i)3-s + (0.0437 − 0.0602i)4-s + (−0.104 − 0.994i)5-s + (−0.473 + 0.153i)6-s + (0.249 + 1.57i)7-s + (−0.150 + 0.947i)8-s + (−0.731 − 0.237i)9-s + (−0.564 − 0.869i)10-s + (−0.277 − 0.960i)11-s + (−0.0253 + 0.0253i)12-s + (−0.116 − 0.228i)13-s + (0.970 + 1.33i)14-s + (−0.0252 + 0.479i)15-s + (0.330 + 1.01i)16-s + (0.212 − 0.416i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.875+0.482i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=((0.875+0.482i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.875+0.482i0.875 + 0.482i
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ55(13,⋅)\chi_{55} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1/2), 0.875+0.482i)(2,\ 55,\ (\ :1/2),\ 0.875 + 0.482i)

Particular Values

L(1)L(1) ≈\approx 1.02006−0.262680i1.02006 - 0.262680i
L(12)L(\frac12) ≈\approx 1.02006−0.262680i1.02006 - 0.262680i
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)
bad5 1+(0.233+2.22i)T 1 + (0.233 + 2.22i)T
11 1+(0.920+3.18i)T 1 + (0.920 + 3.18i)T
good2 1+(−1.30+0.665i)T+(1.17−1.61i)T2 1 + (-1.30 + 0.665i)T + (1.17 - 1.61i)T^{2}
3 1+(0.822+0.130i)T+(2.85+0.927i)T2 1 + (0.822 + 0.130i)T + (2.85 + 0.927i)T^{2}
7 1+(−0.659−4.16i)T+(−6.65+2.16i)T2 1 + (-0.659 - 4.16i)T + (-6.65 + 2.16i)T^{2}
13 1+(0.420+0.824i)T+(−7.64+10.5i)T2 1 + (0.420 + 0.824i)T + (-7.64 + 10.5i)T^{2}
17 1+(−0.875+1.71i)T+(−9.99−13.7i)T2 1 + (-0.875 + 1.71i)T + (-9.99 - 13.7i)T^{2}
19 1+(−4.39+3.19i)T+(5.87−18.0i)T2 1 + (-4.39 + 3.19i)T + (5.87 - 18.0i)T^{2}
23 1+(−1.95−1.95i)T+23iT2 1 + (-1.95 - 1.95i)T + 23iT^{2}
29 1+(0.810+0.588i)T+(8.96+27.5i)T2 1 + (0.810 + 0.588i)T + (8.96 + 27.5i)T^{2}
31 1+(−0.131+0.403i)T+(−25.0−18.2i)T2 1 + (-0.131 + 0.403i)T + (-25.0 - 18.2i)T^{2}
37 1+(4.87−0.771i)T+(35.1−11.4i)T2 1 + (4.87 - 0.771i)T + (35.1 - 11.4i)T^{2}
41 1+(−0.339−0.467i)T+(−12.6+38.9i)T2 1 + (-0.339 - 0.467i)T + (-12.6 + 38.9i)T^{2}
43 1+(5.05−5.05i)T−43iT2 1 + (5.05 - 5.05i)T - 43iT^{2}
47 1+(0.186−1.17i)T+(−44.6−14.5i)T2 1 + (0.186 - 1.17i)T + (-44.6 - 14.5i)T^{2}
53 1+(−8.09+4.12i)T+(31.1−42.8i)T2 1 + (-8.09 + 4.12i)T + (31.1 - 42.8i)T^{2}
59 1+(5.47−7.53i)T+(−18.2−56.1i)T2 1 + (5.47 - 7.53i)T + (-18.2 - 56.1i)T^{2}
61 1+(−7.40+2.40i)T+(49.3−35.8i)T2 1 + (-7.40 + 2.40i)T + (49.3 - 35.8i)T^{2}
67 1+(3.05−3.05i)T−67iT2 1 + (3.05 - 3.05i)T - 67iT^{2}
71 1+(−2.65−8.17i)T+(−57.4+41.7i)T2 1 + (-2.65 - 8.17i)T + (-57.4 + 41.7i)T^{2}
73 1+(−5.39+0.854i)T+(69.4−22.5i)T2 1 + (-5.39 + 0.854i)T + (69.4 - 22.5i)T^{2}
79 1+(−0.705+2.17i)T+(−63.9−46.4i)T2 1 + (-0.705 + 2.17i)T + (-63.9 - 46.4i)T^{2}
83 1+(−1.77−0.902i)T+(48.7+67.1i)T2 1 + (-1.77 - 0.902i)T + (48.7 + 67.1i)T^{2}
89 1+13.9iT−89T2 1 + 13.9iT - 89T^{2}
97 1+(−1.50−2.96i)T+(−57.0+78.4i)T2 1 + (-1.50 - 2.96i)T + (-57.0 + 78.4i)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

−15.13361516037120901191449174769, −13.84518969417529450072254444607, −12.82673055628188539560785855274, −11.80631839672516289205194383634, −11.46215002484084892703654440548, −9.100015532871975858153526619841, −8.305302708803490014798144703748, −5.64145158594276843712038188428, −5.15938250516688489393506567922, −2.98425097363403209050426868035, 3.74359637156638644456182840419, 5.15297678792335216523781939925, 6.63912850779153218725852455218, 7.57057373316167018024527225063, 10.01288863065667166949969980175, 10.75982640677870642877114052021, 12.14083744266960382832170610998, 13.63176426623090742931618503633, 14.24550948013309605291675894759, 15.09922149233619236586153809870

Graph of the ZZ-function along the critical line