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Properties

Label 2-133-133.100-c1-0-3
Degree 22
Conductor 133133
Sign 0.0115−0.999i0.0115 - 0.999i
Analytic cond. 1.062011.06201
Root an. cond. 1.030531.03053
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  + (−0.181 + 1.03i)2-s + (−1.81 + 0.660i)3-s + (0.849 + 0.309i)4-s + (3.78 − 1.37i)5-s + (−0.351 − 1.99i)6-s + (−1.36 + 2.26i)7-s + (−1.52 + 2.63i)8-s + (0.562 − 0.472i)9-s + (0.732 + 4.15i)10-s − 0.481·11-s − 1.74·12-s + (−0.256 − 1.45i)13-s + (−2.08 − 1.82i)14-s + (−5.96 + 5.00i)15-s + (−1.05 − 0.884i)16-s + (3.70 + 3.10i)17-s + ⋯
L(s)  = 1  + (−0.128 + 0.729i)2-s + (−1.04 + 0.381i)3-s + (0.424 + 0.154i)4-s + (1.69 − 0.616i)5-s + (−0.143 − 0.813i)6-s + (−0.516 + 0.856i)7-s + (−0.537 + 0.930i)8-s + (0.187 − 0.157i)9-s + (0.231 + 1.31i)10-s − 0.145·11-s − 0.504·12-s + (−0.0710 − 0.402i)13-s + (−0.557 − 0.486i)14-s + (−1.53 + 1.29i)15-s + (−0.263 − 0.221i)16-s + (0.898 + 0.754i)17-s + ⋯

Functional equation

Λ(s)=(133s/2 ΓC(s) L(s)=((0.0115−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0115 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(133s/2 ΓC(s+1/2) L(s)=((0.0115−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0115 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 133133    =    7⋅197 \cdot 19
Sign: 0.0115−0.999i0.0115 - 0.999i
Analytic conductor: 1.062011.06201
Root analytic conductor: 1.030531.03053
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ133(100,⋅)\chi_{133} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 133, ( :1/2), 0.0115−0.999i)(2,\ 133,\ (\ :1/2),\ 0.0115 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.709120+0.700954i0.709120 + 0.700954i
L(12)L(\frac12) ≈\approx 0.709120+0.700954i0.709120 + 0.700954i
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)
bad7 1+(1.36−2.26i)T 1 + (1.36 - 2.26i)T
19 1+(1.09+4.21i)T 1 + (1.09 + 4.21i)T
good2 1+(0.181−1.03i)T+(−1.87−0.684i)T2 1 + (0.181 - 1.03i)T + (-1.87 - 0.684i)T^{2}
3 1+(1.81−0.660i)T+(2.29−1.92i)T2 1 + (1.81 - 0.660i)T + (2.29 - 1.92i)T^{2}
5 1+(−3.78+1.37i)T+(3.83−3.21i)T2 1 + (-3.78 + 1.37i)T + (3.83 - 3.21i)T^{2}
11 1+0.481T+11T2 1 + 0.481T + 11T^{2}
13 1+(0.256+1.45i)T+(−12.2+4.44i)T2 1 + (0.256 + 1.45i)T + (-12.2 + 4.44i)T^{2}
17 1+(−3.70−3.10i)T+(2.95+16.7i)T2 1 + (-3.70 - 3.10i)T + (2.95 + 16.7i)T^{2}
23 1+(−0.0983−0.558i)T+(−21.6+7.86i)T2 1 + (-0.0983 - 0.558i)T + (-21.6 + 7.86i)T^{2}
29 1+(4.78+1.74i)T+(22.2+18.6i)T2 1 + (4.78 + 1.74i)T + (22.2 + 18.6i)T^{2}
31 1+(−4.12+7.14i)T+(−15.5−26.8i)T2 1 + (-4.12 + 7.14i)T + (-15.5 - 26.8i)T^{2}
37 1+(−0.321+0.556i)T+(−18.5−32.0i)T2 1 + (-0.321 + 0.556i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.283−1.60i)T+(−38.5−14.0i)T2 1 + (0.283 - 1.60i)T + (-38.5 - 14.0i)T^{2}
43 1+(4.52+3.79i)T+(7.46+42.3i)T2 1 + (4.52 + 3.79i)T + (7.46 + 42.3i)T^{2}
47 1+(0.654−0.549i)T+(8.16−46.2i)T2 1 + (0.654 - 0.549i)T + (8.16 - 46.2i)T^{2}
53 1+(−2.64−0.963i)T+(40.6+34.0i)T2 1 + (-2.64 - 0.963i)T + (40.6 + 34.0i)T^{2}
59 1+(−3.30−2.77i)T+(10.2+58.1i)T2 1 + (-3.30 - 2.77i)T + (10.2 + 58.1i)T^{2}
61 1+(2.27+12.9i)T+(−57.3+20.8i)T2 1 + (2.27 + 12.9i)T + (-57.3 + 20.8i)T^{2}
67 1+(−0.946−5.36i)T+(−62.9+22.9i)T2 1 + (-0.946 - 5.36i)T + (-62.9 + 22.9i)T^{2}
71 1+(−6.78−5.69i)T+(12.3+69.9i)T2 1 + (-6.78 - 5.69i)T + (12.3 + 69.9i)T^{2}
73 1+(10.0−3.64i)T+(55.9−46.9i)T2 1 + (10.0 - 3.64i)T + (55.9 - 46.9i)T^{2}
79 1+(−9.04−7.59i)T+(13.7+77.7i)T2 1 + (-9.04 - 7.59i)T + (13.7 + 77.7i)T^{2}
83 1+(7.46+12.9i)T+(−41.5+71.8i)T2 1 + (7.46 + 12.9i)T + (-41.5 + 71.8i)T^{2}
89 1+(−6.39−2.32i)T+(68.1+57.2i)T2 1 + (-6.39 - 2.32i)T + (68.1 + 57.2i)T^{2}
97 1+(13.1−4.78i)T+(74.3−62.3i)T2 1 + (13.1 - 4.78i)T + (74.3 - 62.3i)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

−13.39867509151283495999792386282, −12.52237813749737585157840925683, −11.49505184624107080072568228977, −10.31555891740263418353826002287, −9.407594537764886632129076574598, −8.268890888911534160684817286950, −6.49110964624756277335644775626, −5.77496769564879182526838766465, −5.24702743911382237143202297080, −2.41849859496603097292244072082, 1.45006138188970461248772449061, 3.09604916929483327081351925587, 5.50625001232510902609956550999, 6.40486322185091118388057467123, 7.06134425229595663454507930888, 9.465074899438786998036705956997, 10.23052579274038725599255880673, 10.78749016438028183577731378727, 11.90456803082906850217461763596, 12.80300364166954802938387420019

Graph of the ZZ-function along the critical line