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Properties

Label 2-133-133.100-c1-0-0
Degree 22
Conductor 133133
Sign −0.996−0.0877i-0.996 - 0.0877i
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  + (−0.110 + 0.626i)2-s + (−2.42 + 0.882i)3-s + (1.49 + 0.545i)4-s + (−2.52 + 0.919i)5-s + (−0.284 − 1.61i)6-s + (−1.69 − 2.03i)7-s + (−1.14 + 1.98i)8-s + (2.80 − 2.34i)9-s + (−0.296 − 1.68i)10-s − 3.98·11-s − 4.11·12-s + (0.720 + 4.08i)13-s + (1.45 − 0.836i)14-s + (5.31 − 4.45i)15-s + (1.33 + 1.11i)16-s + (−1.31 − 1.10i)17-s + ⋯
L(s)  = 1  + (−0.0780 + 0.442i)2-s + (−1.39 + 0.509i)3-s + (0.749 + 0.272i)4-s + (−1.12 + 0.411i)5-s + (−0.116 − 0.659i)6-s + (−0.640 − 0.768i)7-s + (−0.404 + 0.700i)8-s + (0.933 − 0.783i)9-s + (−0.0938 − 0.532i)10-s − 1.20·11-s − 1.18·12-s + (0.199 + 1.13i)13-s + (0.390 − 0.223i)14-s + (1.37 − 1.15i)15-s + (0.332 + 0.279i)16-s + (−0.319 − 0.268i)17-s + ⋯

Functional equation

Λ(s)=(133s/2 ΓC(s) L(s)=((−0.996−0.0877i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0877i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(133s/2 ΓC(s+1/2) L(s)=((−0.996−0.0877i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0877i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 133133    =    7⋅197 \cdot 19
Sign: −0.996−0.0877i-0.996 - 0.0877i
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), −0.996−0.0877i)(2,\ 133,\ (\ :1/2),\ -0.996 - 0.0877i)

Particular Values

L(1)L(1) ≈\approx 0.0168430+0.383290i0.0168430 + 0.383290i
L(12)L(\frac12) ≈\approx 0.0168430+0.383290i0.0168430 + 0.383290i
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+(1.69+2.03i)T 1 + (1.69 + 2.03i)T
19 1+(−2.13−3.80i)T 1 + (-2.13 - 3.80i)T
good2 1+(0.110−0.626i)T+(−1.87−0.684i)T2 1 + (0.110 - 0.626i)T + (-1.87 - 0.684i)T^{2}
3 1+(2.42−0.882i)T+(2.29−1.92i)T2 1 + (2.42 - 0.882i)T + (2.29 - 1.92i)T^{2}
5 1+(2.52−0.919i)T+(3.83−3.21i)T2 1 + (2.52 - 0.919i)T + (3.83 - 3.21i)T^{2}
11 1+3.98T+11T2 1 + 3.98T + 11T^{2}
13 1+(−0.720−4.08i)T+(−12.2+4.44i)T2 1 + (-0.720 - 4.08i)T + (-12.2 + 4.44i)T^{2}
17 1+(1.31+1.10i)T+(2.95+16.7i)T2 1 + (1.31 + 1.10i)T + (2.95 + 16.7i)T^{2}
23 1+(−0.354−2.01i)T+(−21.6+7.86i)T2 1 + (-0.354 - 2.01i)T + (-21.6 + 7.86i)T^{2}
29 1+(−8.63−3.14i)T+(22.2+18.6i)T2 1 + (-8.63 - 3.14i)T + (22.2 + 18.6i)T^{2}
31 1+(−1.67+2.89i)T+(−15.5−26.8i)T2 1 + (-1.67 + 2.89i)T + (-15.5 - 26.8i)T^{2}
37 1+(3.43−5.95i)T+(−18.5−32.0i)T2 1 + (3.43 - 5.95i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.629−3.56i)T+(−38.5−14.0i)T2 1 + (0.629 - 3.56i)T + (-38.5 - 14.0i)T^{2}
43 1+(7.59+6.37i)T+(7.46+42.3i)T2 1 + (7.59 + 6.37i)T + (7.46 + 42.3i)T^{2}
47 1+(1.11−0.936i)T+(8.16−46.2i)T2 1 + (1.11 - 0.936i)T + (8.16 - 46.2i)T^{2}
53 1+(−0.133−0.0487i)T+(40.6+34.0i)T2 1 + (-0.133 - 0.0487i)T + (40.6 + 34.0i)T^{2}
59 1+(1.61+1.35i)T+(10.2+58.1i)T2 1 + (1.61 + 1.35i)T + (10.2 + 58.1i)T^{2}
61 1+(1.73+9.85i)T+(−57.3+20.8i)T2 1 + (1.73 + 9.85i)T + (-57.3 + 20.8i)T^{2}
67 1+(−1.22−6.92i)T+(−62.9+22.9i)T2 1 + (-1.22 - 6.92i)T + (-62.9 + 22.9i)T^{2}
71 1+(8.81+7.39i)T+(12.3+69.9i)T2 1 + (8.81 + 7.39i)T + (12.3 + 69.9i)T^{2}
73 1+(−12.5+4.57i)T+(55.9−46.9i)T2 1 + (-12.5 + 4.57i)T + (55.9 - 46.9i)T^{2}
79 1+(2.08+1.75i)T+(13.7+77.7i)T2 1 + (2.08 + 1.75i)T + (13.7 + 77.7i)T^{2}
83 1+(−3.85−6.68i)T+(−41.5+71.8i)T2 1 + (-3.85 - 6.68i)T + (-41.5 + 71.8i)T^{2}
89 1+(−4.11−1.49i)T+(68.1+57.2i)T2 1 + (-4.11 - 1.49i)T + (68.1 + 57.2i)T^{2}
97 1+(10.1−3.69i)T+(74.3−62.3i)T2 1 + (10.1 - 3.69i)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

−13.78153951619973267278148939821, −12.26778450217898075492397718594, −11.61989542257305096258517085500, −10.86530110976043379584402400342, −10.05869522665526268888603883119, −8.098862794822776748885324795213, −7.06505440816487603111965659181, −6.31755340694326939405608598081, −4.85669018064037893079840184673, −3.42315569812514185160298222819, 0.45318034887364618017730142477, 2.90936860842898806370419637011, 5.06686249739886785167651612175, 6.07961062500155336879040819426, 7.15027976872103611343246955203, 8.373909189517295618746886694893, 10.17297321395103779258396751350, 10.94749285640347547076106267377, 11.85078093946280092350143995657, 12.40168373544091742224439772108

Graph of the ZZ-function along the critical line