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Properties

Label 2-197-197.37-c3-0-13
Degree 22
Conductor 197197
Sign −0.712−0.701i-0.712 - 0.701i
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  + (−4.92 + 0.316i)2-s + (4.70 + 6.75i)3-s + (16.2 − 2.09i)4-s + (6.63 + 4.31i)5-s + (−25.3 − 31.7i)6-s + (4.33 + 0.278i)7-s + (−40.7 + 7.92i)8-s + (−14.0 + 38.2i)9-s + (−34.0 − 19.1i)10-s + (8.49 + 28.6i)11-s + (90.7 + 99.9i)12-s + (0.892 + 27.8i)13-s − 21.4·14-s + (2.08 + 65.0i)15-s + (71.2 − 18.6i)16-s + (20.1 − 19.5i)17-s + ⋯
L(s)  = 1  + (−1.74 + 0.111i)2-s + (0.906 + 1.29i)3-s + (2.03 − 0.262i)4-s + (0.593 + 0.386i)5-s + (−1.72 − 2.16i)6-s + (0.234 + 0.0150i)7-s + (−1.79 + 0.350i)8-s + (−0.521 + 1.41i)9-s + (−1.07 − 0.606i)10-s + (0.232 + 0.784i)11-s + (2.18 + 2.40i)12-s + (0.0190 + 0.593i)13-s − 0.410·14-s + (0.0359 + 1.12i)15-s + (1.11 − 0.291i)16-s + (0.287 − 0.278i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((−0.712−0.701i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+3/2) L(s)=((−0.712−0.701i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: −0.712−0.701i-0.712 - 0.701i
Analytic conductor: 11.623311.6233
Root analytic conductor: 3.409303.40930
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ197(37,⋅)\chi_{197} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :3/2), −0.712−0.701i)(2,\ 197,\ (\ :3/2),\ -0.712 - 0.701i)

Particular Values

L(2)L(2) ≈\approx 0.438272+1.06918i0.438272 + 1.06918i
L(12)L(\frac12) ≈\approx 0.438272+1.06918i0.438272 + 1.06918i
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+(−690.+2.67e3i)T 1 + (-690. + 2.67e3i)T
good2 1+(4.92−0.316i)T+(7.93−1.02i)T2 1 + (4.92 - 0.316i)T + (7.93 - 1.02i)T^{2}
3 1+(−4.70−6.75i)T+(−9.32+25.3i)T2 1 + (-4.70 - 6.75i)T + (-9.32 + 25.3i)T^{2}
5 1+(−6.63−4.31i)T+(50.5+114.i)T2 1 + (-6.63 - 4.31i)T + (50.5 + 114. i)T^{2}
7 1+(−4.33−0.278i)T+(340.+43.8i)T2 1 + (-4.33 - 0.278i)T + (340. + 43.8i)T^{2}
11 1+(−8.49−28.6i)T+(−1.11e3+726.i)T2 1 + (-8.49 - 28.6i)T + (-1.11e3 + 726. i)T^{2}
13 1+(−0.892−27.8i)T+(−2.19e3+140.i)T2 1 + (-0.892 - 27.8i)T + (-2.19e3 + 140. i)T^{2}
17 1+(−20.1+19.5i)T+(157.−4.91e3i)T2 1 + (-20.1 + 19.5i)T + (157. - 4.91e3i)T^{2}
19 1+(−80.5+38.8i)T+(4.27e3−5.36e3i)T2 1 + (-80.5 + 38.8i)T + (4.27e3 - 5.36e3i)T^{2}
23 1+(14.0−87.0i)T+(−1.15e4−3.83e3i)T2 1 + (14.0 - 87.0i)T + (-1.15e4 - 3.83e3i)T^{2}
29 1+(−150.−165.i)T+(−2.34e3+2.42e4i)T2 1 + (-150. - 165. i)T + (-2.34e3 + 2.42e4i)T^{2}
31 1+(68.3+27.6i)T+(2.14e4+2.07e4i)T2 1 + (68.3 + 27.6i)T + (2.14e4 + 2.07e4i)T^{2}
37 1+(−27.2−7.14i)T+(4.41e4+2.48e4i)T2 1 + (-27.2 - 7.14i)T + (4.41e4 + 2.48e4i)T^{2}
41 1+(295.−286.i)T+(2.20e3−6.88e4i)T2 1 + (295. - 286. i)T + (2.20e3 - 6.88e4i)T^{2}
43 1+(16.3+54.9i)T+(−6.66e4+4.33e4i)T2 1 + (16.3 + 54.9i)T + (-6.66e4 + 4.33e4i)T^{2}
47 1+(1.17−2.26i)T+(−5.93e4−8.51e4i)T2 1 + (1.17 - 2.26i)T + (-5.93e4 - 8.51e4i)T^{2}
53 1+(−36.6+380.i)T+(−1.46e5−2.84e4i)T2 1 + (-36.6 + 380. i)T + (-1.46e5 - 2.84e4i)T^{2}
59 1+(637.−358.i)T+(1.06e5−1.75e5i)T2 1 + (637. - 358. i)T + (1.06e5 - 1.75e5i)T^{2}
61 1+(−119.+172.i)T+(−7.83e4−2.13e5i)T2 1 + (-119. + 172. i)T + (-7.83e4 - 2.13e5i)T^{2}
67 1+(−379.+726.i)T+(−1.72e5−2.46e5i)T2 1 + (-379. + 726. i)T + (-1.72e5 - 2.46e5i)T^{2}
71 1+(−212.+578.i)T+(−2.72e5−2.32e5i)T2 1 + (-212. + 578. i)T + (-2.72e5 - 2.32e5i)T^{2}
73 1+(534.+140.i)T+(3.38e5+1.90e5i)T2 1 + (534. + 140. i)T + (3.38e5 + 1.90e5i)T^{2}
79 1+(153.−99.9i)T+(1.99e5−4.50e5i)T2 1 + (153. - 99.9i)T + (1.99e5 - 4.50e5i)T^{2}
83 1+(−429.−207.i)T+(3.56e5+4.47e5i)T2 1 + (-429. - 207. i)T + (3.56e5 + 4.47e5i)T^{2}
89 1+(318.−128.i)T+(5.06e5−4.90e5i)T2 1 + (318. - 128. i)T + (5.06e5 - 4.90e5i)T^{2}
97 1+(918.+1.51e3i)T+(−4.22e5+8.09e5i)T2 1 + (918. + 1.51e3i)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

−11.84291739408533154930333740897, −10.84255527364064020063733131102, −9.880428758730323903528596537389, −9.604765524138257422623274854706, −8.747517470304135563672912253322, −7.72419869585069546113211109250, −6.63676587101542698735825895816, −4.85393514394061930547124716087, −3.12094497843744974365688498199, −1.78638302277983958611828827575, 0.831095595483673538537545759871, 1.75055654367911077478542969300, 2.99809799392713960474821572482, 5.88216686646884498162266190261, 7.01097095088246720929052401549, 8.003542865686881627733911132757, 8.465841055198067681370337159849, 9.396991285928697267041549062268, 10.34963145512581520382522243138, 11.58422525750448427240860620655

Graph of the ZZ-function along the critical line