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Properties

Label 2-197-197.16-c3-0-23
Degree 22
Conductor 197197
Sign 0.974−0.223i0.974 - 0.223i
Analytic cond. 11.623311.6233
Root an. cond. 3.409303.40930
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.27 − 0.274i)2-s + (−1.17 + 1.68i)3-s + (10.2 + 1.32i)4-s + (4.43 − 2.88i)5-s + (5.49 − 6.89i)6-s + (4.20 − 0.270i)7-s + (−9.99 − 1.94i)8-s + (7.86 + 21.3i)9-s + (−19.7 + 11.1i)10-s + (−0.707 + 2.38i)11-s + (−14.3 + 15.7i)12-s + (1.85 − 57.9i)13-s − 18.0·14-s + (−0.348 + 10.8i)15-s + (−38.0 − 9.98i)16-s + (−41.5 − 40.2i)17-s + ⋯
L(s)  = 1  + (−1.51 − 0.0971i)2-s + (−0.226 + 0.324i)3-s + (1.28 + 0.165i)4-s + (0.396 − 0.258i)5-s + (0.373 − 0.468i)6-s + (0.227 − 0.0145i)7-s + (−0.441 − 0.0860i)8-s + (0.291 + 0.791i)9-s + (−0.625 + 0.352i)10-s + (−0.0194 + 0.0653i)11-s + (−0.345 + 0.379i)12-s + (0.0396 − 1.23i)13-s − 0.345·14-s + (−0.00600 + 0.187i)15-s + (−0.594 − 0.155i)16-s + (−0.592 − 0.573i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((0.974−0.223i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+3/2) L(s)=((0.974−0.223i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: 0.974−0.223i0.974 - 0.223i
Analytic conductor: 11.623311.6233
Root analytic conductor: 3.409303.40930
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ197(16,⋅)\chi_{197} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :3/2), 0.974−0.223i)(2,\ 197,\ (\ :3/2),\ 0.974 - 0.223i)

Particular Values

L(2)L(2) ≈\approx 0.802365+0.0908597i0.802365 + 0.0908597i
L(12)L(\frac12) ≈\approx 0.802365+0.0908597i0.802365 + 0.0908597i
L(52)L(\frac{5}{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+(−817.+2.64e3i)T 1 + (-817. + 2.64e3i)T
good2 1+(4.27+0.274i)T+(7.93+1.02i)T2 1 + (4.27 + 0.274i)T + (7.93 + 1.02i)T^{2}
3 1+(1.17−1.68i)T+(−9.32−25.3i)T2 1 + (1.17 - 1.68i)T + (-9.32 - 25.3i)T^{2}
5 1+(−4.43+2.88i)T+(50.5−114.i)T2 1 + (-4.43 + 2.88i)T + (50.5 - 114. i)T^{2}
7 1+(−4.20+0.270i)T+(340.−43.8i)T2 1 + (-4.20 + 0.270i)T + (340. - 43.8i)T^{2}
11 1+(0.707−2.38i)T+(−1.11e3−726.i)T2 1 + (0.707 - 2.38i)T + (-1.11e3 - 726. i)T^{2}
13 1+(−1.85+57.9i)T+(−2.19e3−140.i)T2 1 + (-1.85 + 57.9i)T + (-2.19e3 - 140. i)T^{2}
17 1+(41.5+40.2i)T+(157.+4.91e3i)T2 1 + (41.5 + 40.2i)T + (157. + 4.91e3i)T^{2}
19 1+(−12.5−6.04i)T+(4.27e3+5.36e3i)T2 1 + (-12.5 - 6.04i)T + (4.27e3 + 5.36e3i)T^{2}
23 1+(−22.0−136.i)T+(−1.15e4+3.83e3i)T2 1 + (-22.0 - 136. i)T + (-1.15e4 + 3.83e3i)T^{2}
29 1+(−161.+178.i)T+(−2.34e3−2.42e4i)T2 1 + (-161. + 178. i)T + (-2.34e3 - 2.42e4i)T^{2}
31 1+(−78.5+31.8i)T+(2.14e4−2.07e4i)T2 1 + (-78.5 + 31.8i)T + (2.14e4 - 2.07e4i)T^{2}
37 1+(−426.+111.i)T+(4.41e4−2.48e4i)T2 1 + (-426. + 111. i)T + (4.41e4 - 2.48e4i)T^{2}
41 1+(−240.−232.i)T+(2.20e3+6.88e4i)T2 1 + (-240. - 232. i)T + (2.20e3 + 6.88e4i)T^{2}
43 1+(−83.7+282.i)T+(−6.66e4−4.33e4i)T2 1 + (-83.7 + 282. i)T + (-6.66e4 - 4.33e4i)T^{2}
47 1+(−229.−439.i)T+(−5.93e4+8.51e4i)T2 1 + (-229. - 439. i)T + (-5.93e4 + 8.51e4i)T^{2}
53 1+(−11.9−123.i)T+(−1.46e5+2.84e4i)T2 1 + (-11.9 - 123. i)T + (-1.46e5 + 2.84e4i)T^{2}
59 1+(−433.−244.i)T+(1.06e5+1.75e5i)T2 1 + (-433. - 244. i)T + (1.06e5 + 1.75e5i)T^{2}
61 1+(−206.−295.i)T+(−7.83e4+2.13e5i)T2 1 + (-206. - 295. i)T + (-7.83e4 + 2.13e5i)T^{2}
67 1+(391.+749.i)T+(−1.72e5+2.46e5i)T2 1 + (391. + 749. i)T + (-1.72e5 + 2.46e5i)T^{2}
71 1+(−228.−621.i)T+(−2.72e5+2.32e5i)T2 1 + (-228. - 621. i)T + (-2.72e5 + 2.32e5i)T^{2}
73 1+(−489.+128.i)T+(3.38e5−1.90e5i)T2 1 + (-489. + 128. i)T + (3.38e5 - 1.90e5i)T^{2}
79 1+(−310.−202.i)T+(1.99e5+4.50e5i)T2 1 + (-310. - 202. i)T + (1.99e5 + 4.50e5i)T^{2}
83 1+(−1.14e3+552.i)T+(3.56e5−4.47e5i)T2 1 + (-1.14e3 + 552. i)T + (3.56e5 - 4.47e5i)T^{2}
89 1+(901.+364.i)T+(5.06e5+4.90e5i)T2 1 + (901. + 364. i)T + (5.06e5 + 4.90e5i)T^{2}
97 1+(−373.+616.i)T+(−4.22e5−8.09e5i)T2 1 + (-373. + 616. i)T + (-4.22e5 - 8.09e5i)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

−11.52046738347322849901216004540, −10.85585576275673326449153928310, −9.908232074619482298580693028540, −9.331189084350766824018526110092, −8.034389311108303513648236619440, −7.49633825190118363652479689957, −5.84546450144389826597058371097, −4.61832492849827091942251754460, −2.44812261250952228040452896458, −0.931401746792171683583577768046, 0.862602385501125397579068821868, 2.21488236949949676131726119787, 4.37548039908337190100078721356, 6.40605689454880138652231015418, 6.81225765851545200443870099674, 8.172525483773965715713445406346, 9.027185217073459654108432155846, 9.863157556435448077290395058598, 10.79596774457386616771343218377, 11.68535037531631130016237153599

Graph of the ZZ-function along the critical line