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Properties

Label 2-197-197.10-c1-0-15
Degree 22
Conductor 197197
Sign 0.359−0.932i0.359 - 0.932i
Analytic cond. 1.573051.57305
Root an. cond. 1.254211.25421
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.358 − 2.78i)2-s + (−1.19 − 0.441i)3-s + (−5.66 + 1.48i)4-s + (0.0438 − 0.0193i)5-s + (−0.796 + 3.49i)6-s + (−1.37 − 0.177i)7-s + (4.05 + 10.0i)8-s + (−1.04 − 0.887i)9-s + (−0.0696 − 0.114i)10-s + (1.38 + 2.12i)11-s + (7.44 + 0.718i)12-s + (−0.159 − 2.48i)13-s + 3.88i·14-s + (−0.0610 + 0.00392i)15-s + (16.2 − 9.12i)16-s + (−7.15 − 0.229i)17-s + ⋯
L(s)  = 1  + (−0.253 − 1.96i)2-s + (−0.692 − 0.254i)3-s + (−2.83 + 0.742i)4-s + (0.0195 − 0.00867i)5-s + (−0.325 + 1.42i)6-s + (−0.520 − 0.0670i)7-s + (1.43 + 3.54i)8-s + (−0.347 − 0.295i)9-s + (−0.0220 − 0.0363i)10-s + (0.417 + 0.641i)11-s + (2.14 + 0.207i)12-s + (−0.0441 − 0.687i)13-s + 1.03i·14-s + (−0.0157 + 0.00101i)15-s + (4.05 − 2.28i)16-s + (−1.73 − 0.0556i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((0.359−0.932i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+1/2) L(s)=((0.359−0.932i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: 0.359−0.932i0.359 - 0.932i
Analytic conductor: 1.573051.57305
Root analytic conductor: 1.254211.25421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ197(10,⋅)\chi_{197} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :1/2), 0.359−0.932i)(2,\ 197,\ (\ :1/2),\ 0.359 - 0.932i)

Particular Values

L(1)L(1) ≈\approx 0.134404+0.0922043i0.134404 + 0.0922043i
L(12)L(\frac12) ≈\approx 0.134404+0.0922043i0.134404 + 0.0922043i
L(32)L(\frac{3}{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+(12.9−5.32i)T 1 + (12.9 - 5.32i)T
good2 1+(0.358+2.78i)T+(−1.93+0.507i)T2 1 + (0.358 + 2.78i)T + (-1.93 + 0.507i)T^{2}
3 1+(1.19+0.441i)T+(2.28+1.94i)T2 1 + (1.19 + 0.441i)T + (2.28 + 1.94i)T^{2}
5 1+(−0.0438+0.0193i)T+(3.36−3.70i)T2 1 + (-0.0438 + 0.0193i)T + (3.36 - 3.70i)T^{2}
7 1+(1.37+0.177i)T+(6.77+1.77i)T2 1 + (1.37 + 0.177i)T + (6.77 + 1.77i)T^{2}
11 1+(−1.38−2.12i)T+(−4.45+10.0i)T2 1 + (-1.38 - 2.12i)T + (-4.45 + 10.0i)T^{2}
13 1+(0.159+2.48i)T+(−12.8+1.66i)T2 1 + (0.159 + 2.48i)T + (-12.8 + 1.66i)T^{2}
17 1+(7.15+0.229i)T+(16.9+1.08i)T2 1 + (7.15 + 0.229i)T + (16.9 + 1.08i)T^{2}
19 1+(1.48−1.86i)T+(−4.22−18.5i)T2 1 + (1.48 - 1.86i)T + (-4.22 - 18.5i)T^{2}
23 1+(3.60+1.19i)T+(18.4+13.7i)T2 1 + (3.60 + 1.19i)T + (18.4 + 13.7i)T^{2}
29 1+(−0.442+4.58i)T+(−28.4−5.54i)T2 1 + (-0.442 + 4.58i)T + (-28.4 - 5.54i)T^{2}
31 1+(−4.21+4.35i)T+(−0.993−30.9i)T2 1 + (-4.21 + 4.35i)T + (-0.993 - 30.9i)T^{2}
37 1+(5.37+3.02i)T+(19.1+31.6i)T2 1 + (5.37 + 3.02i)T + (19.1 + 31.6i)T^{2}
41 1+(0.165−5.14i)T+(−40.9−2.62i)T2 1 + (0.165 - 5.14i)T + (-40.9 - 2.62i)T^{2}
43 1+(−1.07+0.697i)T+(17.4−39.3i)T2 1 + (-1.07 + 0.697i)T + (17.4 - 39.3i)T^{2}
47 1+(1.07+1.53i)T+(−16.2+44.1i)T2 1 + (1.07 + 1.53i)T + (-16.2 + 44.1i)T^{2}
53 1+(9.14+1.78i)T+(49.1+19.8i)T2 1 + (9.14 + 1.78i)T + (49.1 + 19.8i)T^{2}
59 1+(1.15−1.90i)T+(−27.2−52.3i)T2 1 + (1.15 - 1.90i)T + (-27.2 - 52.3i)T^{2}
61 1+(0.791+2.15i)T+(−46.4+39.5i)T2 1 + (0.791 + 2.15i)T + (-46.4 + 39.5i)T^{2}
67 1+(−10.3+7.25i)T+(23.1−62.8i)T2 1 + (-10.3 + 7.25i)T + (23.1 - 62.8i)T^{2}
71 1+(−3.07+3.61i)T+(−11.3−70.0i)T2 1 + (-3.07 + 3.61i)T + (-11.3 - 70.0i)T^{2}
73 1+(−7.42+13.1i)T+(−37.8−62.4i)T2 1 + (-7.42 + 13.1i)T + (-37.8 - 62.4i)T^{2}
79 1+(−13.4−5.96i)T+(53.1+58.4i)T2 1 + (-13.4 - 5.96i)T + (53.1 + 58.4i)T^{2}
83 1+(10.3+13.0i)T+(−18.4+80.9i)T2 1 + (10.3 + 13.0i)T + (-18.4 + 80.9i)T^{2}
89 1+(3.66+3.77i)T+(−2.85+88.9i)T2 1 + (3.66 + 3.77i)T + (-2.85 + 88.9i)T^{2}
97 1+(0.971−1.86i)T+(−55.4−79.5i)T2 1 + (0.971 - 1.86i)T + (-55.4 - 79.5i)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

−11.63197630377756812412650363282, −10.94835856087882253106003686433, −9.935248242512608958444985619263, −9.195212188349906765809658151826, −8.047776058189067488954312551295, −6.24050978181463555962373972777, −4.76007411059383454465213562678, −3.58225130195631563945093215326, −2.09393820485516562332082652871, −0.16599833274372428335910684646, 4.16315713237469392724847254101, 5.17169034581936715009066666851, 6.34074321263398933467531480803, 6.71768822049406806883177716351, 8.273550813166706405378970464002, 8.935775787543297524645305309580, 9.990463907999834385115959546961, 11.14605868229078626495548021536, 12.61353212316450594530345094716, 13.85381787052922845275614156767

Graph of the ZZ-function along the critical line