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Properties

Label 2-55-55.13-c1-0-0
Degree 22
Conductor 5555
Sign −0.943−0.332i-0.943 - 0.332i
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  + (−0.930 + 0.474i)2-s + (−2.78 − 0.440i)3-s + (−0.533 + 0.734i)4-s + (−1.77 + 1.35i)5-s + (2.79 − 0.909i)6-s + (0.0860 + 0.543i)7-s + (0.475 − 3.00i)8-s + (4.68 + 1.52i)9-s + (1.01 − 2.10i)10-s + (−3.29 − 0.335i)11-s + (1.80 − 1.80i)12-s + (1.47 + 2.89i)13-s + (−0.337 − 0.464i)14-s + (5.54 − 2.99i)15-s + (0.419 + 1.29i)16-s + (−2.57 + 5.04i)17-s + ⋯
L(s)  = 1  + (−0.658 + 0.335i)2-s + (−1.60 − 0.254i)3-s + (−0.266 + 0.367i)4-s + (−0.794 + 0.607i)5-s + (1.14 − 0.371i)6-s + (0.0325 + 0.205i)7-s + (0.168 − 1.06i)8-s + (1.56 + 0.507i)9-s + (0.319 − 0.666i)10-s + (−0.994 − 0.101i)11-s + (0.522 − 0.522i)12-s + (0.408 + 0.801i)13-s + (−0.0902 − 0.124i)14-s + (1.43 − 0.772i)15-s + (0.104 + 0.322i)16-s + (−0.623 + 1.22i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((−0.943−0.332i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=((−0.943−0.332i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: −0.943−0.332i-0.943 - 0.332i
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), −0.943−0.332i)(2,\ 55,\ (\ :1/2),\ -0.943 - 0.332i)

Particular Values

L(1)L(1) ≈\approx 0.0285894+0.167267i0.0285894 + 0.167267i
L(12)L(\frac12) ≈\approx 0.0285894+0.167267i0.0285894 + 0.167267i
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+(1.77−1.35i)T 1 + (1.77 - 1.35i)T
11 1+(3.29+0.335i)T 1 + (3.29 + 0.335i)T
good2 1+(0.930−0.474i)T+(1.17−1.61i)T2 1 + (0.930 - 0.474i)T + (1.17 - 1.61i)T^{2}
3 1+(2.78+0.440i)T+(2.85+0.927i)T2 1 + (2.78 + 0.440i)T + (2.85 + 0.927i)T^{2}
7 1+(−0.0860−0.543i)T+(−6.65+2.16i)T2 1 + (-0.0860 - 0.543i)T + (-6.65 + 2.16i)T^{2}
13 1+(−1.47−2.89i)T+(−7.64+10.5i)T2 1 + (-1.47 - 2.89i)T + (-7.64 + 10.5i)T^{2}
17 1+(2.57−5.04i)T+(−9.99−13.7i)T2 1 + (2.57 - 5.04i)T + (-9.99 - 13.7i)T^{2}
19 1+(1.25−0.914i)T+(5.87−18.0i)T2 1 + (1.25 - 0.914i)T + (5.87 - 18.0i)T^{2}
23 1+(0.803+0.803i)T+23iT2 1 + (0.803 + 0.803i)T + 23iT^{2}
29 1+(3.44+2.50i)T+(8.96+27.5i)T2 1 + (3.44 + 2.50i)T + (8.96 + 27.5i)T^{2}
31 1+(0.509−1.56i)T+(−25.0−18.2i)T2 1 + (0.509 - 1.56i)T + (-25.0 - 18.2i)T^{2}
37 1+(0.945−0.149i)T+(35.1−11.4i)T2 1 + (0.945 - 0.149i)T + (35.1 - 11.4i)T^{2}
41 1+(−5.25−7.23i)T+(−12.6+38.9i)T2 1 + (-5.25 - 7.23i)T + (-12.6 + 38.9i)T^{2}
43 1+(2.55−2.55i)T−43iT2 1 + (2.55 - 2.55i)T - 43iT^{2}
47 1+(−0.636+4.02i)T+(−44.6−14.5i)T2 1 + (-0.636 + 4.02i)T + (-44.6 - 14.5i)T^{2}
53 1+(−6.27+3.19i)T+(31.1−42.8i)T2 1 + (-6.27 + 3.19i)T + (31.1 - 42.8i)T^{2}
59 1+(3.97−5.47i)T+(−18.2−56.1i)T2 1 + (3.97 - 5.47i)T + (-18.2 - 56.1i)T^{2}
61 1+(8.75−2.84i)T+(49.3−35.8i)T2 1 + (8.75 - 2.84i)T + (49.3 - 35.8i)T^{2}
67 1+(2.62−2.62i)T−67iT2 1 + (2.62 - 2.62i)T - 67iT^{2}
71 1+(−2.11−6.51i)T+(−57.4+41.7i)T2 1 + (-2.11 - 6.51i)T + (-57.4 + 41.7i)T^{2}
73 1+(9.96−1.57i)T+(69.4−22.5i)T2 1 + (9.96 - 1.57i)T + (69.4 - 22.5i)T^{2}
79 1+(1.28−3.96i)T+(−63.9−46.4i)T2 1 + (1.28 - 3.96i)T + (-63.9 - 46.4i)T^{2}
83 1+(−10.0−5.14i)T+(48.7+67.1i)T2 1 + (-10.0 - 5.14i)T + (48.7 + 67.1i)T^{2}
89 1−3.64iT−89T2 1 - 3.64iT - 89T^{2}
97 1+(7.56+14.8i)T+(−57.0+78.4i)T2 1 + (7.56 + 14.8i)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

−16.20471740163066784494657306267, −15.21797524310911927942871349716, −13.23307867828626182446021915100, −12.25507812335793588808470534594, −11.19863289618888295913599438237, −10.30110651472893731285097129196, −8.447443235788767794372595750528, −7.23561140345853878943753894305, −6.13509483799179468359776873078, −4.25292693244045137084668134555, 0.37956536279088003648223298822, 4.68785534900789650180266309511, 5.60466936869180153713933009571, 7.56910618015951795584251758372, 9.107038245809173783026941094478, 10.53375170135071189292352337481, 11.08057689761349923192981339155, 12.18804476476780131690585776008, 13.40043347342704795004521847531, 15.35243268677487568655446903069

Graph of the ZZ-function along the critical line