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Properties

Label 2-197-197.16-c3-0-20
Degree 22
Conductor 197197
Sign −0.208−0.977i-0.208 - 0.977i
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  + (−1.84 − 0.118i)2-s + (−3.90 + 5.60i)3-s + (−4.53 − 0.584i)4-s + (−3.25 + 2.12i)5-s + (7.88 − 9.89i)6-s + (29.1 − 1.87i)7-s + (22.8 + 4.45i)8-s + (−6.79 − 18.4i)9-s + (6.27 − 3.53i)10-s + (5.84 − 19.6i)11-s + (20.9 − 23.1i)12-s + (0.543 − 16.9i)13-s − 54.1·14-s + (0.851 − 26.5i)15-s + (−6.34 − 1.66i)16-s + (16.8 + 16.3i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.0419i)2-s + (−0.752 + 1.07i)3-s + (−0.566 − 0.0730i)4-s + (−0.291 + 0.189i)5-s + (0.536 − 0.673i)6-s + (1.57 − 0.101i)7-s + (1.00 + 0.196i)8-s + (−0.251 − 0.683i)9-s + (0.198 − 0.111i)10-s + (0.160 − 0.539i)11-s + (0.504 − 0.555i)12-s + (0.0115 − 0.361i)13-s − 1.03·14-s + (0.0146 − 0.456i)15-s + (−0.0991 − 0.0260i)16-s + (0.240 + 0.233i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((−0.208−0.977i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+3/2) L(s)=((−0.208−0.977i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: −0.208−0.977i-0.208 - 0.977i
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), −0.208−0.977i)(2,\ 197,\ (\ :3/2),\ -0.208 - 0.977i)

Particular Values

L(2)L(2) ≈\approx 0.523773+0.647350i0.523773 + 0.647350i
L(12)L(\frac12) ≈\approx 0.523773+0.647350i0.523773 + 0.647350i
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+(2.62e3+857.i)T 1 + (2.62e3 + 857. i)T
good2 1+(1.84+0.118i)T+(7.93+1.02i)T2 1 + (1.84 + 0.118i)T + (7.93 + 1.02i)T^{2}
3 1+(3.90−5.60i)T+(−9.32−25.3i)T2 1 + (3.90 - 5.60i)T + (-9.32 - 25.3i)T^{2}
5 1+(3.25−2.12i)T+(50.5−114.i)T2 1 + (3.25 - 2.12i)T + (50.5 - 114. i)T^{2}
7 1+(−29.1+1.87i)T+(340.−43.8i)T2 1 + (-29.1 + 1.87i)T + (340. - 43.8i)T^{2}
11 1+(−5.84+19.6i)T+(−1.11e3−726.i)T2 1 + (-5.84 + 19.6i)T + (-1.11e3 - 726. i)T^{2}
13 1+(−0.543+16.9i)T+(−2.19e3−140.i)T2 1 + (-0.543 + 16.9i)T + (-2.19e3 - 140. i)T^{2}
17 1+(−16.8−16.3i)T+(157.+4.91e3i)T2 1 + (-16.8 - 16.3i)T + (157. + 4.91e3i)T^{2}
19 1+(−78.1−37.6i)T+(4.27e3+5.36e3i)T2 1 + (-78.1 - 37.6i)T + (4.27e3 + 5.36e3i)T^{2}
23 1+(−7.88−48.7i)T+(−1.15e4+3.83e3i)T2 1 + (-7.88 - 48.7i)T + (-1.15e4 + 3.83e3i)T^{2}
29 1+(43.7−48.2i)T+(−2.34e3−2.42e4i)T2 1 + (43.7 - 48.2i)T + (-2.34e3 - 2.42e4i)T^{2}
31 1+(−222.+90.1i)T+(2.14e4−2.07e4i)T2 1 + (-222. + 90.1i)T + (2.14e4 - 2.07e4i)T^{2}
37 1+(102.−26.9i)T+(4.41e4−2.48e4i)T2 1 + (102. - 26.9i)T + (4.41e4 - 2.48e4i)T^{2}
41 1+(−112.−109.i)T+(2.20e3+6.88e4i)T2 1 + (-112. - 109. i)T + (2.20e3 + 6.88e4i)T^{2}
43 1+(87.6−295.i)T+(−6.66e4−4.33e4i)T2 1 + (87.6 - 295. i)T + (-6.66e4 - 4.33e4i)T^{2}
47 1+(126.+243.i)T+(−5.93e4+8.51e4i)T2 1 + (126. + 243. i)T + (-5.93e4 + 8.51e4i)T^{2}
53 1+(−16.0−166.i)T+(−1.46e5+2.84e4i)T2 1 + (-16.0 - 166. i)T + (-1.46e5 + 2.84e4i)T^{2}
59 1+(−18.2−10.2i)T+(1.06e5+1.75e5i)T2 1 + (-18.2 - 10.2i)T + (1.06e5 + 1.75e5i)T^{2}
61 1+(−157.−225.i)T+(−7.83e4+2.13e5i)T2 1 + (-157. - 225. i)T + (-7.83e4 + 2.13e5i)T^{2}
67 1+(−132.−253.i)T+(−1.72e5+2.46e5i)T2 1 + (-132. - 253. i)T + (-1.72e5 + 2.46e5i)T^{2}
71 1+(−250.−680.i)T+(−2.72e5+2.32e5i)T2 1 + (-250. - 680. i)T + (-2.72e5 + 2.32e5i)T^{2}
73 1+(−580.+152.i)T+(3.38e5−1.90e5i)T2 1 + (-580. + 152. i)T + (3.38e5 - 1.90e5i)T^{2}
79 1+(629.+409.i)T+(1.99e5+4.50e5i)T2 1 + (629. + 409. i)T + (1.99e5 + 4.50e5i)T^{2}
83 1+(181.−87.6i)T+(3.56e5−4.47e5i)T2 1 + (181. - 87.6i)T + (3.56e5 - 4.47e5i)T^{2}
89 1+(51.8+20.9i)T+(5.06e5+4.90e5i)T2 1 + (51.8 + 20.9i)T + (5.06e5 + 4.90e5i)T^{2}
97 1+(152.−251.i)T+(−4.22e5−8.09e5i)T2 1 + (152. - 251. 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

−11.69919663331466956299424335989, −11.19609456410636164659018466379, −10.34833437513532974635568674322, −9.539433428870727892337606354943, −8.326826234322660830510646538920, −7.63511034122059763048917691650, −5.59073334406214201669389453786, −4.87238052760350544500372840681, −3.81810786221146059918818431904, −1.19533846375108161082384430358, 0.64737227994097425712550961505, 1.74391961825812210884572232479, 4.40740262851093328390138295849, 5.30831537430726747893056894794, 6.89750068905353348869486981698, 7.76503337706282301303255708135, 8.473856808207040397847572203714, 9.688382657418484939686701464996, 10.96064628755246284497915381507, 11.83309635680555653509490611071

Graph of the ZZ-function along the critical line