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Properties

Label 2-197-197.36-c3-0-34
Degree 22
Conductor 197197
Sign −0.668+0.743i-0.668 + 0.743i
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  + (−4.04 + 1.94i)2-s + (−1.44 − 0.694i)3-s + (7.55 − 9.47i)4-s + (1.68 + 2.11i)5-s + 7.18·6-s + (0.423 + 0.204i)7-s + (−4.10 + 17.9i)8-s + (−15.2 − 19.1i)9-s + (−10.9 − 5.25i)10-s + (25.8 − 12.4i)11-s + (−17.4 + 8.41i)12-s + (12.3 + 53.9i)13-s − 2.10·14-s + (−0.961 − 4.21i)15-s + (3.14 + 13.7i)16-s + (25.8 + 32.4i)17-s + ⋯
L(s)  = 1  + (−1.42 + 0.688i)2-s + (−0.277 − 0.133i)3-s + (0.944 − 1.18i)4-s + (0.150 + 0.188i)5-s + 0.488·6-s + (0.0228 + 0.0110i)7-s + (−0.181 + 0.795i)8-s + (−0.564 − 0.707i)9-s + (−0.344 − 0.166i)10-s + (0.708 − 0.341i)11-s + (−0.420 + 0.202i)12-s + (0.262 + 1.15i)13-s − 0.0402·14-s + (−0.0165 − 0.0725i)15-s + (0.0491 + 0.215i)16-s + (0.368 + 0.462i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((−0.668+0.743i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+3/2) L(s)=((−0.668+0.743i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: −0.668+0.743i-0.668 + 0.743i
Analytic conductor: 11.623311.6233
Root analytic conductor: 3.409303.40930
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ197(36,⋅)\chi_{197} (36, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :3/2), −0.668+0.743i)(2,\ 197,\ (\ :3/2),\ -0.668 + 0.743i)

Particular Values

L(2)L(2) ≈\approx 0.0301651−0.0677167i0.0301651 - 0.0677167i
L(12)L(\frac12) ≈\approx 0.0301651−0.0677167i0.0301651 - 0.0677167i
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+(2.33e3−1.47e3i)T 1 + (2.33e3 - 1.47e3i)T
good2 1+(4.04−1.94i)T+(4.98−6.25i)T2 1 + (4.04 - 1.94i)T + (4.98 - 6.25i)T^{2}
3 1+(1.44+0.694i)T+(16.8+21.1i)T2 1 + (1.44 + 0.694i)T + (16.8 + 21.1i)T^{2}
5 1+(−1.68−2.11i)T+(−27.8+121.i)T2 1 + (-1.68 - 2.11i)T + (-27.8 + 121. i)T^{2}
7 1+(−0.423−0.204i)T+(213.+268.i)T2 1 + (-0.423 - 0.204i)T + (213. + 268. i)T^{2}
11 1+(−25.8+12.4i)T+(829.−1.04e3i)T2 1 + (-25.8 + 12.4i)T + (829. - 1.04e3i)T^{2}
13 1+(−12.3−53.9i)T+(−1.97e3+953.i)T2 1 + (-12.3 - 53.9i)T + (-1.97e3 + 953. i)T^{2}
17 1+(−25.8−32.4i)T+(−1.09e3+4.78e3i)T2 1 + (-25.8 - 32.4i)T + (-1.09e3 + 4.78e3i)T^{2}
19 1+71.6T+6.85e3T2 1 + 71.6T + 6.85e3T^{2}
23 1+(136.−65.8i)T+(7.58e3−9.51e3i)T2 1 + (136. - 65.8i)T + (7.58e3 - 9.51e3i)T^{2}
29 1+(192.−92.7i)T+(1.52e4−1.90e4i)T2 1 + (192. - 92.7i)T + (1.52e4 - 1.90e4i)T^{2}
31 1+(−235.+113.i)T+(1.85e4−2.32e4i)T2 1 + (-235. + 113. i)T + (1.85e4 - 2.32e4i)T^{2}
37 1+(−88.8+389.i)T+(−4.56e4−2.19e4i)T2 1 + (-88.8 + 389. i)T + (-4.56e4 - 2.19e4i)T^{2}
41 1+(202.+253.i)T+(−1.53e4+6.71e4i)T2 1 + (202. + 253. i)T + (-1.53e4 + 6.71e4i)T^{2}
43 1+(228.−109.i)T+(4.95e4−6.21e4i)T2 1 + (228. - 109. i)T + (4.95e4 - 6.21e4i)T^{2}
47 1+(−22.5+98.5i)T+(−9.35e4−4.50e4i)T2 1 + (-22.5 + 98.5i)T + (-9.35e4 - 4.50e4i)T^{2}
53 1+(410.−515.i)T+(−3.31e4−1.45e5i)T2 1 + (410. - 515. i)T + (-3.31e4 - 1.45e5i)T^{2}
59 1+(772.−372.i)T+(1.28e5−1.60e5i)T2 1 + (772. - 372. i)T + (1.28e5 - 1.60e5i)T^{2}
61 1+(416.−200.i)T+(1.41e5−1.77e5i)T2 1 + (416. - 200. i)T + (1.41e5 - 1.77e5i)T^{2}
67 1+(−199.+872.i)T+(−2.70e5−1.30e5i)T2 1 + (-199. + 872. i)T + (-2.70e5 - 1.30e5i)T^{2}
71 1+(246.+309.i)T+(−7.96e4+3.48e5i)T2 1 + (246. + 309. i)T + (-7.96e4 + 3.48e5i)T^{2}
73 1+(49.1−215.i)T+(−3.50e5−1.68e5i)T2 1 + (49.1 - 215. i)T + (-3.50e5 - 1.68e5i)T^{2}
79 1+(4.25−5.33i)T+(−1.09e5−4.80e5i)T2 1 + (4.25 - 5.33i)T + (-1.09e5 - 4.80e5i)T^{2}
83 1+55.5T+5.71e5T2 1 + 55.5T + 5.71e5T^{2}
89 1+(−74.5−35.9i)T+(4.39e5+5.51e5i)T2 1 + (-74.5 - 35.9i)T + (4.39e5 + 5.51e5i)T^{2}
97 1+(−209.−262.i)T+(−2.03e5+8.89e5i)T2 1 + (-209. - 262. i)T + (-2.03e5 + 8.89e5i)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

−11.43376823008017191311140534995, −10.47183330653194763588765257764, −9.364169410314021717271423719157, −8.783328264324981775042779051236, −7.70679807737497551123151331520, −6.43464639680630103087370345496, −6.08555693330257012206935273691, −3.90898467406336373355008711555, −1.67846081766722274153740996755, −0.05445054953283462447787568397, 1.55053531776499581254319723181, 2.99895031442001807879555586786, 4.90425347753627730012103357287, 6.31720000751922983485888563702, 7.889964889001539099954381177640, 8.411386765502455783710797944587, 9.644905844276152245550279999244, 10.28381413968529779191723615225, 11.22757838471522135346368149558, 11.92392795803608369315634142861

Graph of the ZZ-function along the critical line