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Properties

Label 2-197-197.37-c3-0-25
Degree 22
Conductor 197197
Sign −0.697+0.716i-0.697 + 0.716i
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  + (−2.99 + 0.192i)2-s + (−1.86 − 2.67i)3-s + (0.972 − 0.125i)4-s + (10.2 + 6.68i)5-s + (6.09 + 7.63i)6-s + (−26.7 − 1.71i)7-s + (20.6 − 4.02i)8-s + (5.65 − 15.3i)9-s + (−32.0 − 18.0i)10-s + (5.46 + 18.4i)11-s + (−2.15 − 2.36i)12-s + (1.98 + 61.7i)13-s + 80.3·14-s + (−1.28 − 39.9i)15-s + (−68.5 + 17.9i)16-s + (89.0 − 86.2i)17-s + ⋯
L(s)  = 1  + (−1.05 + 0.0678i)2-s + (−0.358 − 0.514i)3-s + (0.121 − 0.0156i)4-s + (0.918 + 0.598i)5-s + (0.414 + 0.519i)6-s + (−1.44 − 0.0927i)7-s + (0.912 − 0.177i)8-s + (0.209 − 0.568i)9-s + (−1.01 − 0.570i)10-s + (0.149 + 0.505i)11-s + (−0.0517 − 0.0569i)12-s + (0.0422 + 1.31i)13-s + 1.53·14-s + (−0.0220 − 0.687i)15-s + (−1.07 + 0.280i)16-s + (1.27 − 1.23i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((−0.697+0.716i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+3/2) L(s)=((−0.697+0.716i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: −0.697+0.716i-0.697 + 0.716i
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), −0.697+0.716i)(2,\ 197,\ (\ :3/2),\ -0.697 + 0.716i)

Particular Values

L(2)L(2) ≈\approx 0.135584−0.321140i0.135584 - 0.321140i
L(12)L(\frac12) ≈\approx 0.135584−0.321140i0.135584 - 0.321140i
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+(2.68e3+673.i)T 1 + (2.68e3 + 673. i)T
good2 1+(2.99−0.192i)T+(7.93−1.02i)T2 1 + (2.99 - 0.192i)T + (7.93 - 1.02i)T^{2}
3 1+(1.86+2.67i)T+(−9.32+25.3i)T2 1 + (1.86 + 2.67i)T + (-9.32 + 25.3i)T^{2}
5 1+(−10.2−6.68i)T+(50.5+114.i)T2 1 + (-10.2 - 6.68i)T + (50.5 + 114. i)T^{2}
7 1+(26.7+1.71i)T+(340.+43.8i)T2 1 + (26.7 + 1.71i)T + (340. + 43.8i)T^{2}
11 1+(−5.46−18.4i)T+(−1.11e3+726.i)T2 1 + (-5.46 - 18.4i)T + (-1.11e3 + 726. i)T^{2}
13 1+(−1.98−61.7i)T+(−2.19e3+140.i)T2 1 + (-1.98 - 61.7i)T + (-2.19e3 + 140. i)T^{2}
17 1+(−89.0+86.2i)T+(157.−4.91e3i)T2 1 + (-89.0 + 86.2i)T + (157. - 4.91e3i)T^{2}
19 1+(−10.6+5.12i)T+(4.27e3−5.36e3i)T2 1 + (-10.6 + 5.12i)T + (4.27e3 - 5.36e3i)T^{2}
23 1+(3.56−22.0i)T+(−1.15e4−3.83e3i)T2 1 + (3.56 - 22.0i)T + (-1.15e4 - 3.83e3i)T^{2}
29 1+(83.5+91.9i)T+(−2.34e3+2.42e4i)T2 1 + (83.5 + 91.9i)T + (-2.34e3 + 2.42e4i)T^{2}
31 1+(236.+95.9i)T+(2.14e4+2.07e4i)T2 1 + (236. + 95.9i)T + (2.14e4 + 2.07e4i)T^{2}
37 1+(356.+93.3i)T+(4.41e4+2.48e4i)T2 1 + (356. + 93.3i)T + (4.41e4 + 2.48e4i)T^{2}
41 1+(−12.1+11.7i)T+(2.20e3−6.88e4i)T2 1 + (-12.1 + 11.7i)T + (2.20e3 - 6.88e4i)T^{2}
43 1+(105.+355.i)T+(−6.66e4+4.33e4i)T2 1 + (105. + 355. i)T + (-6.66e4 + 4.33e4i)T^{2}
47 1+(−142.+273.i)T+(−5.93e4−8.51e4i)T2 1 + (-142. + 273. i)T + (-5.93e4 - 8.51e4i)T^{2}
53 1+(−27.5+285.i)T+(−1.46e5−2.84e4i)T2 1 + (-27.5 + 285. i)T + (-1.46e5 - 2.84e4i)T^{2}
59 1+(325.−183.i)T+(1.06e5−1.75e5i)T2 1 + (325. - 183. i)T + (1.06e5 - 1.75e5i)T^{2}
61 1+(−46.4+66.5i)T+(−7.83e4−2.13e5i)T2 1 + (-46.4 + 66.5i)T + (-7.83e4 - 2.13e5i)T^{2}
67 1+(−179.+344.i)T+(−1.72e5−2.46e5i)T2 1 + (-179. + 344. i)T + (-1.72e5 - 2.46e5i)T^{2}
71 1+(226.−616.i)T+(−2.72e5−2.32e5i)T2 1 + (226. - 616. i)T + (-2.72e5 - 2.32e5i)T^{2}
73 1+(−433.−113.i)T+(3.38e5+1.90e5i)T2 1 + (-433. - 113. i)T + (3.38e5 + 1.90e5i)T^{2}
79 1+(359.−233.i)T+(1.99e5−4.50e5i)T2 1 + (359. - 233. i)T + (1.99e5 - 4.50e5i)T^{2}
83 1+(904.+435.i)T+(3.56e5+4.47e5i)T2 1 + (904. + 435. i)T + (3.56e5 + 4.47e5i)T^{2}
89 1+(229.−92.7i)T+(5.06e5−4.90e5i)T2 1 + (229. - 92.7i)T + (5.06e5 - 4.90e5i)T^{2}
97 1+(583.+961.i)T+(−4.22e5+8.09e5i)T2 1 + (583. + 961. 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

−11.62298222669880403486369282170, −10.18566520040050508376634890525, −9.631655524694708313253033894363, −9.116142453755987135515726826446, −7.14879009245963657699518661938, −6.94778488218938882851026272378, −5.68927489714540572492288390593, −3.70208587695999805483577572925, −1.87227510886124834125119487997, −0.24032590404654647741636954460, 1.39873031729724135635394512094, 3.44574378400896675152683966479, 5.22242006648101789842259604985, 5.96439294331689137763231890660, 7.59969747698291144933660058552, 8.701829546251404988974442694755, 9.597291663076891723954581082468, 10.21830839531786094790117007549, 10.78826916638229708156396968988, 12.65584416402048342817134164262

Graph of the ZZ-function along the critical line