↑

Properties

Label 2-55-55.2-c1-0-2
Degree 22
Conductor 5555
Sign 0.661+0.749i0.661 + 0.749i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.261 − 0.513i)2-s + (0.120 − 0.760i)3-s + (0.980 − 1.34i)4-s + (−1.76 + 1.36i)5-s + (−0.421 + 0.136i)6-s + (1.17 − 0.186i)7-s + (−2.08 − 0.330i)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 + (−5.41 + 2.75i)13-s + (−0.403 − 0.555i)14-s + (0.825 + 1.51i)15-s + (−0.655 − 2.01i)16-s + (0.330 + 0.168i)17-s + ⋯
L(s)  = 1  + (−0.184 − 0.362i)2-s + (0.0695 − 0.438i)3-s + (0.490 − 0.674i)4-s + (−0.791 + 0.611i)5-s + (−0.172 + 0.0559i)6-s + (0.445 − 0.0705i)7-s + (−0.737 − 0.116i)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 + (−1.50 + 0.765i)13-s + (−0.107 − 0.148i)14-s + (0.213 + 0.389i)15-s + (−0.163 − 0.504i)16-s + (0.0801 + 0.0408i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.661+0.749i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=((0.661+0.749i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.661+0.749i0.661 + 0.749i
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ55(2,⋅)\chi_{55} (2, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1/2), 0.661+0.749i)(2,\ 55,\ (\ :1/2),\ 0.661 + 0.749i)

Particular Values

L(1)L(1) ≈\approx 0.738277−0.333224i0.738277 - 0.333224i
L(12)L(\frac12) ≈\approx 0.738277−0.333224i0.738277 - 0.333224i
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+(1.76−1.36i)T 1 + (1.76 - 1.36i)T
11 1+(0.502−3.27i)T 1 + (0.502 - 3.27i)T
good2 1+(0.261+0.513i)T+(−1.17+1.61i)T2 1 + (0.261 + 0.513i)T + (-1.17 + 1.61i)T^{2}
3 1+(−0.120+0.760i)T+(−2.85−0.927i)T2 1 + (-0.120 + 0.760i)T + (-2.85 - 0.927i)T^{2}
7 1+(−1.17+0.186i)T+(6.65−2.16i)T2 1 + (-1.17 + 0.186i)T + (6.65 - 2.16i)T^{2}
13 1+(5.41−2.75i)T+(7.64−10.5i)T2 1 + (5.41 - 2.75i)T + (7.64 - 10.5i)T^{2}
17 1+(−0.330−0.168i)T+(9.99+13.7i)T2 1 + (-0.330 - 0.168i)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+(0.465+2.93i)T+(−35.1+11.4i)T2 1 + (0.465 + 2.93i)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+(1.26+0.199i)T+(44.6+14.5i)T2 1 + (1.26 + 0.199i)T + (44.6 + 14.5i)T^{2}
53 1+(−0.185−0.363i)T+(−31.1+42.8i)T2 1 + (-0.185 - 0.363i)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+(−0.715−4.51i)T+(−69.4+22.5i)T2 1 + (-0.715 - 4.51i)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+(−5.16+10.1i)T+(−48.7−67.1i)T2 1 + (-5.16 + 10.1i)T + (-48.7 - 67.1i)T^{2}
89 1−8.04iT−89T2 1 - 8.04iT - 89T^{2}
97 1+(−7.52+3.83i)T+(57.0−78.4i)T2 1 + (-7.52 + 3.83i)T + (57.0 - 78.4i)T^{2}
show more
show less
   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

−15.00597945975132628554837847930, −14.40296267572224139518533038774, −12.60226283157882052521678759037, −11.72375437909449330904455988577, −10.61283937000209686574001720657, −9.575765427438229473013973658590, −7.53483875237760741378546355654, −6.87470734013728367665563832871, −4.68686299969765933047277472462, −2.23701452163949610681131906161, 3.45552171887150784266093300837, 5.15726334648729579152720252335, 7.25032399552082226870441340822, 8.107058564439821617868113746097, 9.381735464886320909418381026164, 11.02380155404627652560807891873, 12.09851512397631091060124262723, 12.99352743719574080358343410173, 14.88101611477977938100491300300, 15.55196528483753144229930700224

Graph of the ZZ-function along the critical line