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Properties

Label 2-55-55.13-c1-0-2
Degree 22
Conductor 5555
Sign 0.877−0.479i0.877 - 0.479i
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  + (−0.513 + 0.261i)2-s + (0.760 + 0.120i)3-s + (−0.980 + 1.34i)4-s + (2.23 + 0.0653i)5-s + (−0.421 + 0.136i)6-s + (−0.186 − 1.17i)7-s + (0.330 − 2.08i)8-s + (−2.28 − 0.743i)9-s + (−1.16 + 0.550i)10-s + (−0.502 + 3.27i)11-s + (−0.908 + 0.908i)12-s + (−2.75 − 5.41i)13-s + (0.403 + 0.555i)14-s + (1.69 + 0.318i)15-s + (−0.655 − 2.01i)16-s + (0.168 − 0.330i)17-s + ⋯
L(s)  = 1  + (−0.362 + 0.184i)2-s + (0.438 + 0.0695i)3-s + (−0.490 + 0.674i)4-s + (0.999 + 0.0292i)5-s + (−0.172 + 0.0559i)6-s + (−0.0705 − 0.445i)7-s + (0.116 − 0.737i)8-s + (−0.763 − 0.247i)9-s + (−0.368 + 0.174i)10-s + (−0.151 + 0.988i)11-s + (−0.262 + 0.262i)12-s + (−0.765 − 1.50i)13-s + (0.107 + 0.148i)14-s + (0.436 + 0.0823i)15-s + (−0.163 − 0.504i)16-s + (0.0408 − 0.0801i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.877−0.479i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=((0.877−0.479i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.877−0.479i0.877 - 0.479i
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), 0.877−0.479i)(2,\ 55,\ (\ :1/2),\ 0.877 - 0.479i)

Particular Values

L(1)L(1) ≈\approx 0.767489+0.195867i0.767489 + 0.195867i
L(12)L(\frac12) ≈\approx 0.767489+0.195867i0.767489 + 0.195867i
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+(−2.23−0.0653i)T 1 + (-2.23 - 0.0653i)T
11 1+(0.502−3.27i)T 1 + (0.502 - 3.27i)T
good2 1+(0.513−0.261i)T+(1.17−1.61i)T2 1 + (0.513 - 0.261i)T + (1.17 - 1.61i)T^{2}
3 1+(−0.760−0.120i)T+(2.85+0.927i)T2 1 + (-0.760 - 0.120i)T + (2.85 + 0.927i)T^{2}
7 1+(0.186+1.17i)T+(−6.65+2.16i)T2 1 + (0.186 + 1.17i)T + (-6.65 + 2.16i)T^{2}
13 1+(2.75+5.41i)T+(−7.64+10.5i)T2 1 + (2.75 + 5.41i)T + (-7.64 + 10.5i)T^{2}
17 1+(−0.168+0.330i)T+(−9.99−13.7i)T2 1 + (-0.168 + 0.330i)T + (-9.99 - 13.7i)T^{2}
19 1+(1.11−0.809i)T+(5.87−18.0i)T2 1 + (1.11 - 0.809i)T + (5.87 - 18.0i)T^{2}
23 1+(−3.48−3.48i)T+23iT2 1 + (-3.48 - 3.48i)T + 23iT^{2}
29 1+(−4.95−3.60i)T+(8.96+27.5i)T2 1 + (-4.95 - 3.60i)T + (8.96 + 27.5i)T^{2}
31 1+(0.764−2.35i)T+(−25.0−18.2i)T2 1 + (0.764 - 2.35i)T + (-25.0 - 18.2i)T^{2}
37 1+(2.93−0.465i)T+(35.1−11.4i)T2 1 + (2.93 - 0.465i)T + (35.1 - 11.4i)T^{2}
41 1+(3.60+4.95i)T+(−12.6+38.9i)T2 1 + (3.60 + 4.95i)T + (-12.6 + 38.9i)T^{2}
43 1+(6.75−6.75i)T−43iT2 1 + (6.75 - 6.75i)T - 43iT^{2}
47 1+(0.199−1.26i)T+(−44.6−14.5i)T2 1 + (0.199 - 1.26i)T + (-44.6 - 14.5i)T^{2}
53 1+(0.363−0.185i)T+(31.1−42.8i)T2 1 + (0.363 - 0.185i)T + (31.1 - 42.8i)T^{2}
59 1+(0.110−0.151i)T+(−18.2−56.1i)T2 1 + (0.110 - 0.151i)T + (-18.2 - 56.1i)T^{2}
61 1+(0.649−0.211i)T+(49.3−35.8i)T2 1 + (0.649 - 0.211i)T + (49.3 - 35.8i)T^{2}
67 1+(−7.14+7.14i)T−67iT2 1 + (-7.14 + 7.14i)T - 67iT^{2}
71 1+(−0.319−0.983i)T+(−57.4+41.7i)T2 1 + (-0.319 - 0.983i)T + (-57.4 + 41.7i)T^{2}
73 1+(4.51−0.715i)T+(69.4−22.5i)T2 1 + (4.51 - 0.715i)T + (69.4 - 22.5i)T^{2}
79 1+(3.59−11.0i)T+(−63.9−46.4i)T2 1 + (3.59 - 11.0i)T + (-63.9 - 46.4i)T^{2}
83 1+(−10.1−5.16i)T+(48.7+67.1i)T2 1 + (-10.1 - 5.16i)T + (48.7 + 67.1i)T^{2}
89 1+8.04iT−89T2 1 + 8.04iT - 89T^{2}
97 1+(3.83+7.52i)T+(−57.0+78.4i)T2 1 + (3.83 + 7.52i)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

−15.33665710256822567967228509308, −14.25666724935322634046424420542, −13.21267599197319831998639991783, −12.35018312988895599195886852198, −10.35507176867738924062927099307, −9.484992858112086209732425645312, −8.310391446766748147252574655422, −7.06164199751535296062940296713, −5.14512056402110959283836608049, −3.06776956194270685861345794717, 2.33585851796872187937261904254, 5.07695155006679707522361662820, 6.33697028836350616658316391568, 8.574569381147008223696247611439, 9.192513447439731936749690187327, 10.38967898743068641538431323318, 11.63697805215827628259738498649, 13.40463427409499280812466599298, 14.08178260826312105661652051769, 14.86637735715550441357073181691

Graph of the ZZ-function along the critical line