↑

Properties

Label 2-197-197.6-c3-0-28
Degree 22
Conductor 197197
Sign 0.992−0.121i0.992 - 0.121i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.93 + 1.12i)2-s + (8.18 + 1.86i)3-s + (15.8 − 7.62i)4-s + (3.18 − 6.62i)5-s − 42.4·6-s + (1.15 − 5.04i)7-s + (−37.8 + 30.2i)8-s + (39.1 + 18.8i)9-s + (−8.27 + 36.2i)10-s + (25.8 − 5.91i)11-s + (143. − 32.8i)12-s + (23.1 + 18.4i)13-s + 26.1i·14-s + (38.4 − 48.2i)15-s + (65.1 − 81.6i)16-s + (7.25 − 15.0i)17-s + ⋯
L(s)  = 1  + (−1.74 + 0.397i)2-s + (1.57 + 0.359i)3-s + (1.97 − 0.953i)4-s + (0.285 − 0.592i)5-s − 2.88·6-s + (0.0621 − 0.272i)7-s + (−1.67 + 1.33i)8-s + (1.45 + 0.698i)9-s + (−0.261 + 1.14i)10-s + (0.709 − 0.162i)11-s + (3.46 − 0.789i)12-s + (0.494 + 0.394i)13-s + 0.499i·14-s + (0.662 − 0.830i)15-s + (1.01 − 1.27i)16-s + (0.103 − 0.214i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((0.992−0.121i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+3/2) L(s)=((0.992−0.121i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: 0.992−0.121i0.992 - 0.121i
Analytic conductor: 11.623311.6233
Root analytic conductor: 3.409303.40930
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ197(6,⋅)\chi_{197} (6, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :3/2), 0.992−0.121i)(2,\ 197,\ (\ :3/2),\ 0.992 - 0.121i)

Particular Values

L(2)L(2) ≈\approx 1.49776+0.0912270i1.49776 + 0.0912270i
L(12)L(\frac12) ≈\approx 1.49776+0.0912270i1.49776 + 0.0912270i
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+(−1.20e3+2.48e3i)T 1 + (-1.20e3 + 2.48e3i)T
good2 1+(4.93−1.12i)T+(7.20−3.47i)T2 1 + (4.93 - 1.12i)T + (7.20 - 3.47i)T^{2}
3 1+(−8.18−1.86i)T+(24.3+11.7i)T2 1 + (-8.18 - 1.86i)T + (24.3 + 11.7i)T^{2}
5 1+(−3.18+6.62i)T+(−77.9−97.7i)T2 1 + (-3.18 + 6.62i)T + (-77.9 - 97.7i)T^{2}
7 1+(−1.15+5.04i)T+(−309.−148.i)T2 1 + (-1.15 + 5.04i)T + (-309. - 148. i)T^{2}
11 1+(−25.8+5.91i)T+(1.19e3−577.i)T2 1 + (-25.8 + 5.91i)T + (1.19e3 - 577. i)T^{2}
13 1+(−23.1−18.4i)T+(488.+2.14e3i)T2 1 + (-23.1 - 18.4i)T + (488. + 2.14e3i)T^{2}
17 1+(−7.25+15.0i)T+(−3.06e3−3.84e3i)T2 1 + (-7.25 + 15.0i)T + (-3.06e3 - 3.84e3i)T^{2}
19 1+31.0T+6.85e3T2 1 + 31.0T + 6.85e3T^{2}
23 1+(32.9+144.i)T+(−1.09e4+5.27e3i)T2 1 + (32.9 + 144. i)T + (-1.09e4 + 5.27e3i)T^{2}
29 1+(29.7+130.i)T+(−2.19e4+1.05e4i)T2 1 + (29.7 + 130. i)T + (-2.19e4 + 1.05e4i)T^{2}
31 1+(−159.+36.4i)T+(2.68e4−1.29e4i)T2 1 + (-159. + 36.4i)T + (2.68e4 - 1.29e4i)T^{2}
37 1+(−221.−278.i)T+(−1.12e4+4.93e4i)T2 1 + (-221. - 278. i)T + (-1.12e4 + 4.93e4i)T^{2}
41 1+(143.+69.0i)T+(4.29e4+5.38e4i)T2 1 + (143. + 69.0i)T + (4.29e4 + 5.38e4i)T^{2}
43 1+(98.5+431.i)T+(−7.16e4+3.44e4i)T2 1 + (98.5 + 431. i)T + (-7.16e4 + 3.44e4i)T^{2}
47 1+(−219.−274.i)T+(−2.31e4+1.01e5i)T2 1 + (-219. - 274. i)T + (-2.31e4 + 1.01e5i)T^{2}
53 1+(68.0−32.7i)T+(9.28e4−1.16e5i)T2 1 + (68.0 - 32.7i)T + (9.28e4 - 1.16e5i)T^{2}
59 1+(33.1+145.i)T+(−1.85e5+8.91e4i)T2 1 + (33.1 + 145. i)T + (-1.85e5 + 8.91e4i)T^{2}
61 1+(−157.−691.i)T+(−2.04e5+9.84e4i)T2 1 + (-157. - 691. i)T + (-2.04e5 + 9.84e4i)T^{2}
67 1+(−150.+119.i)T+(6.69e4−2.93e5i)T2 1 + (-150. + 119. i)T + (6.69e4 - 2.93e5i)T^{2}
71 1+(437.−907.i)T+(−2.23e5−2.79e5i)T2 1 + (437. - 907. i)T + (-2.23e5 - 2.79e5i)T^{2}
73 1+(210.−167.i)T+(8.65e4−3.79e5i)T2 1 + (210. - 167. i)T + (8.65e4 - 3.79e5i)T^{2}
79 1+(247.+514.i)T+(−3.07e5+3.85e5i)T2 1 + (247. + 514. i)T + (-3.07e5 + 3.85e5i)T^{2}
83 1−514.T+5.71e5T2 1 - 514.T + 5.71e5T^{2}
89 1+(485.+110.i)T+(6.35e5+3.05e5i)T2 1 + (485. + 110. i)T + (6.35e5 + 3.05e5i)T^{2}
97 1+(227.+109.i)T+(5.69e5+7.13e5i)T2 1 + (227. + 109. i)T + (5.69e5 + 7.13e5i)T^{2}
show more
show less
   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

−11.72454668741035447229818321789, −10.43097515896401252467534085293, −9.699999301225574096509998717335, −8.795507384635870428050639806527, −8.524686411977234450587784809591, −7.44438836776553567463246363619, −6.30204523362256341060305685467, −4.20639443014894254396601111054, −2.46674325980225438445175787331, −1.14784043162500858584377555147, 1.40539188849730914082470022787, 2.46182118980919778434657154491, 3.50440472035969963347307938561, 6.42677085592603641620651229632, 7.43639556328173620258730094000, 8.241231784020657768672553746337, 9.007269124703987559904544977609, 9.719660793677155453050109780603, 10.64563161655910059596888560471, 11.73973628920147073891146824845

Graph of the ZZ-function along the critical line