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Properties

Label 2-197-197.158-c1-0-10
Degree 22
Conductor 197197
Sign 0.0305+0.999i0.0305 + 0.999i
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  + (−0.375 + 2.32i)2-s + (0.220 − 0.242i)3-s + (−3.36 − 1.11i)4-s + (−4.10 − 0.529i)5-s + (0.480 + 0.602i)6-s + (−0.687 − 4.25i)7-s + (1.68 − 3.23i)8-s + (0.277 + 2.87i)9-s + (2.77 − 9.35i)10-s + (−2.04 + 0.131i)11-s + (−1.01 + 0.570i)12-s + (0.928 − 0.790i)13-s + 10.1·14-s + (−1.03 + 0.878i)15-s + (1.20 + 0.896i)16-s + (−1.03 + 2.80i)17-s + ⋯
L(s)  = 1  + (−0.265 + 1.64i)2-s + (0.127 − 0.139i)3-s + (−1.68 − 0.559i)4-s + (−1.83 − 0.236i)5-s + (0.196 + 0.246i)6-s + (−0.259 − 1.60i)7-s + (0.596 − 1.14i)8-s + (0.0925 + 0.959i)9-s + (0.877 − 2.95i)10-s + (−0.615 + 0.0395i)11-s + (−0.292 + 0.164i)12-s + (0.257 − 0.219i)13-s + 2.71·14-s + (−0.266 + 0.226i)15-s + (0.300 + 0.224i)16-s + (−0.249 + 0.679i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((0.0305+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0305 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+1/2) L(s)=((0.0305+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0305 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: 0.0305+0.999i0.0305 + 0.999i
Analytic conductor: 1.573051.57305
Root analytic conductor: 1.254211.25421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ197(158,⋅)\chi_{197} (158, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :1/2), 0.0305+0.999i)(2,\ 197,\ (\ :1/2),\ 0.0305 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.0146190−0.0141793i0.0146190 - 0.0141793i
L(12)L(\frac12) ≈\approx 0.0146190−0.0141793i0.0146190 - 0.0141793i
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+(13.9−1.87i)T 1 + (13.9 - 1.87i)T
good2 1+(0.375−2.32i)T+(−1.89−0.630i)T2 1 + (0.375 - 2.32i)T + (-1.89 - 0.630i)T^{2}
3 1+(−0.220+0.242i)T+(−0.288−2.98i)T2 1 + (-0.220 + 0.242i)T + (-0.288 - 2.98i)T^{2}
5 1+(4.10+0.529i)T+(4.83+1.26i)T2 1 + (4.10 + 0.529i)T + (4.83 + 1.26i)T^{2}
7 1+(0.687+4.25i)T+(−6.64+2.20i)T2 1 + (0.687 + 4.25i)T + (-6.64 + 2.20i)T^{2}
11 1+(2.04−0.131i)T+(10.9−1.40i)T2 1 + (2.04 - 0.131i)T + (10.9 - 1.40i)T^{2}
13 1+(−0.928+0.790i)T+(2.07−12.8i)T2 1 + (-0.928 + 0.790i)T + (2.07 - 12.8i)T^{2}
17 1+(1.03−2.80i)T+(−12.9−11.0i)T2 1 + (1.03 - 2.80i)T + (-12.9 - 11.0i)T^{2}
19 1+(6.57−3.16i)T+(11.8−14.8i)T2 1 + (6.57 - 3.16i)T + (11.8 - 14.8i)T^{2}
23 1+(4.12+1.66i)T+(16.5+16.0i)T2 1 + (4.12 + 1.66i)T + (16.5 + 16.0i)T^{2}
29 1+(0.450−0.253i)T+(15.0−24.7i)T2 1 + (0.450 - 0.253i)T + (15.0 - 24.7i)T^{2}
31 1+(−2.44+3.50i)T+(−10.7−29.0i)T2 1 + (-2.44 + 3.50i)T + (-10.7 - 29.0i)T^{2}
37 1+(−1.53+1.14i)T+(10.5−35.4i)T2 1 + (-1.53 + 1.14i)T + (10.5 - 35.4i)T^{2}
41 1+(−1.18+3.21i)T+(−31.2−26.5i)T2 1 + (-1.18 + 3.21i)T + (-31.2 - 26.5i)T^{2}
43 1+(6.11−0.392i)T+(42.6−5.49i)T2 1 + (6.11 - 0.392i)T + (42.6 - 5.49i)T^{2}
47 1+(0.0126+0.0284i)T+(−31.5+34.7i)T2 1 + (0.0126 + 0.0284i)T + (-31.5 + 34.7i)T^{2}
53 1+(7.03−11.5i)T+(−24.5−46.9i)T2 1 + (7.03 - 11.5i)T + (-24.5 - 46.9i)T^{2}
59 1+(1.46+4.94i)T+(−49.4+32.1i)T2 1 + (1.46 + 4.94i)T + (-49.4 + 32.1i)T^{2}
61 1+(−2.74−3.02i)T+(−5.85+60.7i)T2 1 + (-2.74 - 3.02i)T + (-5.85 + 60.7i)T^{2}
67 1+(0.00351+0.00793i)T+(−45.0+49.5i)T2 1 + (0.00351 + 0.00793i)T + (-45.0 + 49.5i)T^{2}
71 1+(0.893+9.25i)T+(−69.6+13.5i)T2 1 + (0.893 + 9.25i)T + (-69.6 + 13.5i)T^{2}
73 1+(−9.09+6.78i)T+(20.7−69.9i)T2 1 + (-9.09 + 6.78i)T + (20.7 - 69.9i)T^{2}
79 1+(17.1−2.21i)T+(76.4−20.0i)T2 1 + (17.1 - 2.21i)T + (76.4 - 20.0i)T^{2}
83 1+(−6.92−3.33i)T+(51.7+64.8i)T2 1 + (-6.92 - 3.33i)T + (51.7 + 64.8i)T^{2}
89 1+(−1.78−2.55i)T+(−30.7+83.5i)T2 1 + (-1.78 - 2.55i)T + (-30.7 + 83.5i)T^{2}
97 1+(5.45+3.55i)T+(39.2+88.6i)T2 1 + (5.45 + 3.55i)T + (39.2 + 88.6i)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

−12.59992474258052895559138752096, −11.01381883333191991884855649530, −10.30572743515295201315581078950, −8.487430465425033291527351075917, −7.911150327228490146284148590812, −7.46147651321945821740280285904, −6.35776559310999878439271471091, −4.64250446989531780945416706767, −3.97088074403857039398515156673, −0.01864561737587065103422834263, 2.59526075004943595176512177735, 3.52307744550557911905265389325, 4.62747082538380709781393052763, 6.60600646553523494730281445215, 8.320709441374417311315262973876, 8.817883462324472848064119976699, 9.897823267320403774522912330747, 11.17572081898359195418656695967, 11.67088390061573607818943649855, 12.33466201450057084952961427872

Graph of the ZZ-function along the critical line