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Properties

Label 2-550-11.5-c1-0-13
Degree 22
Conductor 550550
Sign 0.0219+0.999i0.0219 + 0.999i
Analytic cond. 4.391774.39177
Root an. cond. 2.095652.09565
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  + (−0.309 − 0.951i)2-s + (0.5 + 0.363i)3-s + (−0.809 + 0.587i)4-s + (0.190 − 0.587i)6-s + (1.80 − 1.31i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 2.48i)9-s + (−1.23 − 3.07i)11-s − 0.618·12-s + (1.30 + 4.02i)13-s + (−1.80 − 1.31i)14-s + (0.309 − 0.951i)16-s + (0.809 − 2.48i)17-s + (−2.11 + 1.53i)18-s + (−0.309 − 0.224i)19-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.288 + 0.209i)3-s + (−0.404 + 0.293i)4-s + (0.0779 − 0.239i)6-s + (0.683 − 0.496i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.829i)9-s + (−0.372 − 0.927i)11-s − 0.178·12-s + (0.363 + 1.11i)13-s + (−0.483 − 0.351i)14-s + (0.0772 − 0.237i)16-s + (0.196 − 0.603i)17-s + (−0.499 + 0.362i)18-s + (−0.0708 − 0.0515i)19-s + ⋯

Functional equation

Λ(s)=(550s/2 ΓC(s) L(s)=((0.0219+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(550s/2 ΓC(s+1/2) L(s)=((0.0219+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 550550    =    2⋅52⋅112 \cdot 5^{2} \cdot 11
Sign: 0.0219+0.999i0.0219 + 0.999i
Analytic conductor: 4.391774.39177
Root analytic conductor: 2.095652.09565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ550(401,⋅)\chi_{550} (401, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 550, ( :1/2), 0.0219+0.999i)(2,\ 550,\ (\ :1/2),\ 0.0219 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.985172−0.963747i0.985172 - 0.963747i
L(12)L(\frac12) ≈\approx 0.985172−0.963747i0.985172 - 0.963747i
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)
bad2 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
5 1 1
11 1+(1.23+3.07i)T 1 + (1.23 + 3.07i)T
good3 1+(−0.5−0.363i)T+(0.927+2.85i)T2 1 + (-0.5 - 0.363i)T + (0.927 + 2.85i)T^{2}
7 1+(−1.80+1.31i)T+(2.16−6.65i)T2 1 + (-1.80 + 1.31i)T + (2.16 - 6.65i)T^{2}
13 1+(−1.30−4.02i)T+(−10.5+7.64i)T2 1 + (-1.30 - 4.02i)T + (-10.5 + 7.64i)T^{2}
17 1+(−0.809+2.48i)T+(−13.7−9.99i)T2 1 + (-0.809 + 2.48i)T + (-13.7 - 9.99i)T^{2}
19 1+(0.309+0.224i)T+(5.87+18.0i)T2 1 + (0.309 + 0.224i)T + (5.87 + 18.0i)T^{2}
23 1−2.85T+23T2 1 - 2.85T + 23T^{2}
29 1+(−7.66+5.56i)T+(8.96−27.5i)T2 1 + (-7.66 + 5.56i)T + (8.96 - 27.5i)T^{2}
31 1+(1.78+5.48i)T+(−25.0+18.2i)T2 1 + (1.78 + 5.48i)T + (-25.0 + 18.2i)T^{2}
37 1+(−1.73+1.26i)T+(11.4−35.1i)T2 1 + (-1.73 + 1.26i)T + (11.4 - 35.1i)T^{2}
41 1+(7.85+5.70i)T+(12.6+38.9i)T2 1 + (7.85 + 5.70i)T + (12.6 + 38.9i)T^{2}
43 1+0.527T+43T2 1 + 0.527T + 43T^{2}
47 1+(−3.42−2.48i)T+(14.5+44.6i)T2 1 + (-3.42 - 2.48i)T + (14.5 + 44.6i)T^{2}
53 1+(−1.11−3.44i)T+(−42.8+31.1i)T2 1 + (-1.11 - 3.44i)T + (-42.8 + 31.1i)T^{2}
59 1+(−2.04+1.48i)T+(18.2−56.1i)T2 1 + (-2.04 + 1.48i)T + (18.2 - 56.1i)T^{2}
61 1+(0.690−2.12i)T+(−49.3−35.8i)T2 1 + (0.690 - 2.12i)T + (-49.3 - 35.8i)T^{2}
67 1−6.32T+67T2 1 - 6.32T + 67T^{2}
71 1+(4.57−14.0i)T+(−57.4−41.7i)T2 1 + (4.57 - 14.0i)T + (-57.4 - 41.7i)T^{2}
73 1+(12.0−8.73i)T+(22.5−69.4i)T2 1 + (12.0 - 8.73i)T + (22.5 - 69.4i)T^{2}
79 1+(−1.83−5.65i)T+(−63.9+46.4i)T2 1 + (-1.83 - 5.65i)T + (-63.9 + 46.4i)T^{2}
83 1+(−2.78+8.55i)T+(−67.1−48.7i)T2 1 + (-2.78 + 8.55i)T + (-67.1 - 48.7i)T^{2}
89 1−9.18T+89T2 1 - 9.18T + 89T^{2}
97 1+(−3.70−11.4i)T+(−78.4+57.0i)T2 1 + (-3.70 - 11.4i)T + (-78.4 + 57.0i)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

−10.63676265253664432663881169786, −9.694459773198906708735256592731, −8.867212636620179763009773907126, −8.225266920821470604339491579037, −7.08599327818047503478115493529, −5.92266679071401456802796133195, −4.59240063397054813432271120616, −3.71650288545644717929278058225, −2.56456482855910133228178817823, −0.909834619166162361229114892992, 1.66898707992793268170819857843, 3.07826092333088880951295018187, 4.84592775826124543848570205482, 5.31602000236144683602024260028, 6.58848741665435749858409045822, 7.63046634759462610059548962024, 8.259295774337207122779362967891, 8.871959323442526062742066184177, 10.25790449642380175191092203002, 10.67131162618513741580798975928

Graph of the ZZ-function along the critical line