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Properties

Label 2-197-197.100-c1-0-0
Degree 22
Conductor 197197
Sign −0.826−0.563i-0.826 - 0.563i
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  + (0.661 − 0.173i)2-s + (−0.901 − 0.767i)3-s + (−1.33 + 0.751i)4-s + (−2.05 + 2.25i)5-s + (−0.729 − 0.351i)6-s + (−2.04 − 0.535i)7-s + (−1.73 + 1.68i)8-s + (−0.255 − 1.57i)9-s + (−0.965 + 1.85i)10-s + (−0.971 + 2.19i)11-s + (1.78 + 0.346i)12-s + (−0.311 + 0.0401i)13-s − 1.44·14-s + (3.58 − 0.462i)15-s + (0.732 − 1.20i)16-s + (−1.70 − 0.109i)17-s + ⋯
L(s)  = 1  + (0.467 − 0.122i)2-s + (−0.520 − 0.442i)3-s + (−0.667 + 0.375i)4-s + (−0.917 + 1.01i)5-s + (−0.297 − 0.143i)6-s + (−0.771 − 0.202i)7-s + (−0.613 + 0.594i)8-s + (−0.0850 − 0.526i)9-s + (−0.305 + 0.585i)10-s + (−0.292 + 0.661i)11-s + (0.513 + 0.100i)12-s + (−0.0863 + 0.0111i)13-s − 0.385·14-s + (0.925 − 0.119i)15-s + (0.183 − 0.302i)16-s + (−0.413 − 0.0265i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((−0.826−0.563i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+1/2) L(s)=((−0.826−0.563i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: −0.826−0.563i-0.826 - 0.563i
Analytic conductor: 1.573051.57305
Root analytic conductor: 1.254211.25421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ197(100,⋅)\chi_{197} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :1/2), −0.826−0.563i)(2,\ 197,\ (\ :1/2),\ -0.826 - 0.563i)

Particular Values

L(1)L(1) ≈\approx 0.0852926+0.276580i0.0852926 + 0.276580i
L(12)L(\frac12) ≈\approx 0.0852926+0.276580i0.0852926 + 0.276580i
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+(−14.0−0.430i)T 1 + (-14.0 - 0.430i)T
good2 1+(−0.661+0.173i)T+(1.74−0.981i)T2 1 + (-0.661 + 0.173i)T + (1.74 - 0.981i)T^{2}
3 1+(0.901+0.767i)T+(0.478+2.96i)T2 1 + (0.901 + 0.767i)T + (0.478 + 2.96i)T^{2}
5 1+(2.05−2.25i)T+(−0.480−4.97i)T2 1 + (2.05 - 2.25i)T + (-0.480 - 4.97i)T^{2}
7 1+(2.04+0.535i)T+(6.09+3.43i)T2 1 + (2.04 + 0.535i)T + (6.09 + 3.43i)T^{2}
11 1+(0.971−2.19i)T+(−7.39−8.14i)T2 1 + (0.971 - 2.19i)T + (-7.39 - 8.14i)T^{2}
13 1+(0.311−0.0401i)T+(12.5−3.29i)T2 1 + (0.311 - 0.0401i)T + (12.5 - 3.29i)T^{2}
17 1+(1.70+0.109i)T+(16.8+2.17i)T2 1 + (1.70 + 0.109i)T + (16.8 + 2.17i)T^{2}
19 1+(−0.307−1.34i)T+(−17.1+8.24i)T2 1 + (-0.307 - 1.34i)T + (-17.1 + 8.24i)T^{2}
23 1+(−6.44−4.80i)T+(6.54+22.0i)T2 1 + (-6.44 - 4.80i)T + (6.54 + 22.0i)T^{2}
29 1+(4.06+0.792i)T+(26.8+10.8i)T2 1 + (4.06 + 0.792i)T + (26.8 + 10.8i)T^{2}
31 1+(0.0946+2.95i)T+(−30.9+1.98i)T2 1 + (0.0946 + 2.95i)T + (-30.9 + 1.98i)T^{2}
37 1+(2.84+4.70i)T+(−17.1+32.8i)T2 1 + (2.84 + 4.70i)T + (-17.1 + 32.8i)T^{2}
41 1+(11.4+0.733i)T+(40.6+5.24i)T2 1 + (11.4 + 0.733i)T + (40.6 + 5.24i)T^{2}
43 1+(3.85−8.69i)T+(−28.9−31.8i)T2 1 + (3.85 - 8.69i)T + (-28.9 - 31.8i)T^{2}
47 1+(−1.83+4.97i)T+(−35.7−30.4i)T2 1 + (-1.83 + 4.97i)T + (-35.7 - 30.4i)T^{2}
53 1+(−8.97−3.63i)T+(38.0+36.8i)T2 1 + (-8.97 - 3.63i)T + (38.0 + 36.8i)T^{2}
59 1+(−6.78−12.9i)T+(−33.7+48.3i)T2 1 + (-6.78 - 12.9i)T + (-33.7 + 48.3i)T^{2}
61 1+(5.87−4.99i)T+(9.73−60.2i)T2 1 + (5.87 - 4.99i)T + (9.73 - 60.2i)T^{2}
67 1+(0.845−2.29i)T+(−51.0−43.4i)T2 1 + (0.845 - 2.29i)T + (-51.0 - 43.4i)T^{2}
71 1+(−0.959−5.93i)T+(−67.3+22.3i)T2 1 + (-0.959 - 5.93i)T + (-67.3 + 22.3i)T^{2}
73 1+(−3.22−5.31i)T+(−33.7+64.7i)T2 1 + (-3.22 - 5.31i)T + (-33.7 + 64.7i)T^{2}
79 1+(5.14+5.66i)T+(−7.58+78.6i)T2 1 + (5.14 + 5.66i)T + (-7.58 + 78.6i)T^{2}
83 1+(−1.26+5.54i)T+(−74.7−36.0i)T2 1 + (-1.26 + 5.54i)T + (-74.7 - 36.0i)T^{2}
89 1+(0.158−4.93i)T+(−88.8−5.70i)T2 1 + (0.158 - 4.93i)T + (-88.8 - 5.70i)T^{2}
97 1+(1.87+2.69i)T+(−33.5+91.0i)T2 1 + (1.87 + 2.69i)T + (-33.5 + 91.0i)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

−12.88154755961236706192103350215, −11.92237889525726199121395035208, −11.33274171971045186844288966830, −10.02869008665277034453016045995, −8.923589071844206392712191202490, −7.47010480586380823157746547702, −6.85148834577901455609457908254, −5.49430576660087265013413273551, −3.96429378268650754452004150908, −3.09958653089239399915351920742, 0.23042069631894926541971987883, 3.48349484763176800868797192571, 4.76918179801761432276918029564, 5.27730771255927516199225419268, 6.64552138901980804846252045160, 8.332219960689758822571303289905, 8.998137072545795229724821595430, 10.17922725202754483530225311298, 11.18169673959266012661028185898, 12.28278208203515960950940148298

Graph of the ZZ-function along the critical line