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Properties

Label 2-197-197.10-c1-0-4
Degree 22
Conductor 197197
Sign 0.311−0.950i0.311 - 0.950i
Analytic cond. 1.573051.57305
Root an. cond. 1.254211.25421
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  + (0.235 + 1.82i)2-s + (−1.41 − 0.519i)3-s + (−1.35 + 0.354i)4-s + (3.07 − 1.36i)5-s + (0.616 − 2.70i)6-s + (1.72 + 0.221i)7-s + (0.417 + 1.03i)8-s + (−0.564 − 0.480i)9-s + (3.21 + 5.30i)10-s + (1.93 + 2.97i)11-s + (2.08 + 0.201i)12-s + (−0.00162 − 0.0253i)13-s + 3.19i·14-s + (−5.04 + 0.324i)15-s + (−4.21 + 2.37i)16-s + (−7.09 − 0.227i)17-s + ⋯
L(s)  = 1  + (0.166 + 1.29i)2-s + (−0.814 − 0.299i)3-s + (−0.675 + 0.177i)4-s + (1.37 − 0.609i)5-s + (0.251 − 1.10i)6-s + (0.650 + 0.0839i)7-s + (0.147 + 0.364i)8-s + (−0.188 − 0.160i)9-s + (1.01 + 1.67i)10-s + (0.583 + 0.896i)11-s + (0.603 + 0.0581i)12-s + (−0.000451 − 0.00702i)13-s + 0.855i·14-s + (−1.30 + 0.0836i)15-s + (−1.05 + 0.594i)16-s + (−1.71 − 0.0551i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((0.311−0.950i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+1/2) L(s)=((0.311−0.950i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: 0.311−0.950i0.311 - 0.950i
Analytic conductor: 1.573051.57305
Root analytic conductor: 1.254211.25421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ197(10,⋅)\chi_{197} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :1/2), 0.311−0.950i)(2,\ 197,\ (\ :1/2),\ 0.311 - 0.950i)

Particular Values

L(1)L(1) ≈\approx 1.06383+0.770739i1.06383 + 0.770739i
L(12)L(\frac12) ≈\approx 1.06383+0.770739i1.06383 + 0.770739i
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)
bad197 1+(12.6−5.98i)T 1 + (12.6 - 5.98i)T
good2 1+(−0.235−1.82i)T+(−1.93+0.507i)T2 1 + (-0.235 - 1.82i)T + (-1.93 + 0.507i)T^{2}
3 1+(1.41+0.519i)T+(2.28+1.94i)T2 1 + (1.41 + 0.519i)T + (2.28 + 1.94i)T^{2}
5 1+(−3.07+1.36i)T+(3.36−3.70i)T2 1 + (-3.07 + 1.36i)T + (3.36 - 3.70i)T^{2}
7 1+(−1.72−0.221i)T+(6.77+1.77i)T2 1 + (-1.72 - 0.221i)T + (6.77 + 1.77i)T^{2}
11 1+(−1.93−2.97i)T+(−4.45+10.0i)T2 1 + (-1.93 - 2.97i)T + (-4.45 + 10.0i)T^{2}
13 1+(0.00162+0.0253i)T+(−12.8+1.66i)T2 1 + (0.00162 + 0.0253i)T + (-12.8 + 1.66i)T^{2}
17 1+(7.09+0.227i)T+(16.9+1.08i)T2 1 + (7.09 + 0.227i)T + (16.9 + 1.08i)T^{2}
19 1+(−3.73+4.67i)T+(−4.22−18.5i)T2 1 + (-3.73 + 4.67i)T + (-4.22 - 18.5i)T^{2}
23 1+(−5.39−1.79i)T+(18.4+13.7i)T2 1 + (-5.39 - 1.79i)T + (18.4 + 13.7i)T^{2}
29 1+(0.0981−1.01i)T+(−28.4−5.54i)T2 1 + (0.0981 - 1.01i)T + (-28.4 - 5.54i)T^{2}
31 1+(1.74−1.80i)T+(−0.993−30.9i)T2 1 + (1.74 - 1.80i)T + (-0.993 - 30.9i)T^{2}
37 1+(6.97+3.92i)T+(19.1+31.6i)T2 1 + (6.97 + 3.92i)T + (19.1 + 31.6i)T^{2}
41 1+(−0.195+6.09i)T+(−40.9−2.62i)T2 1 + (-0.195 + 6.09i)T + (-40.9 - 2.62i)T^{2}
43 1+(6.91−4.49i)T+(17.4−39.3i)T2 1 + (6.91 - 4.49i)T + (17.4 - 39.3i)T^{2}
47 1+(5.29+7.59i)T+(−16.2+44.1i)T2 1 + (5.29 + 7.59i)T + (-16.2 + 44.1i)T^{2}
53 1+(−5.96−1.16i)T+(49.1+19.8i)T2 1 + (-5.96 - 1.16i)T + (49.1 + 19.8i)T^{2}
59 1+(−3.02+4.98i)T+(−27.2−52.3i)T2 1 + (-3.02 + 4.98i)T + (-27.2 - 52.3i)T^{2}
61 1+(3.68+10.0i)T+(−46.4+39.5i)T2 1 + (3.68 + 10.0i)T + (-46.4 + 39.5i)T^{2}
67 1+(4.49−3.13i)T+(23.1−62.8i)T2 1 + (4.49 - 3.13i)T + (23.1 - 62.8i)T^{2}
71 1+(0.356−0.418i)T+(−11.3−70.0i)T2 1 + (0.356 - 0.418i)T + (-11.3 - 70.0i)T^{2}
73 1+(0.188−0.335i)T+(−37.8−62.4i)T2 1 + (0.188 - 0.335i)T + (-37.8 - 62.4i)T^{2}
79 1+(8.92+3.95i)T+(53.1+58.4i)T2 1 + (8.92 + 3.95i)T + (53.1 + 58.4i)T^{2}
83 1+(−4.68−5.87i)T+(−18.4+80.9i)T2 1 + (-4.68 - 5.87i)T + (-18.4 + 80.9i)T^{2}
89 1+(−10.0−10.3i)T+(−2.85+88.9i)T2 1 + (-10.0 - 10.3i)T + (-2.85 + 88.9i)T^{2}
97 1+(6.52−12.5i)T+(−55.4−79.5i)T2 1 + (6.52 - 12.5i)T + (-55.4 - 79.5i)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

−12.98898522442670859662859145297, −11.72687359740397187530913583098, −10.89130074745008415022714084283, −9.301325388063228906234367801892, −8.730766700139651653971568981372, −7.02431702289028269807153184472, −6.56642304658558824631153584752, −5.27863188720498924270942535929, −4.95182289186391788104520258440, −1.85344062525715946578111910953, 1.65761570676836233845974465177, 3.00584958978506210387618352015, 4.63804081789556674424634714277, 5.82195291008100164718684349334, 6.78625702537659103695293060341, 8.711331469766409523037130258187, 9.839393884797584752071288735534, 10.65920265021098484618917286317, 11.20141042976455117126448851036, 11.87235148719255420835614704780

Graph of the ZZ-function along the critical line