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Properties

Label 2-197-197.10-c1-0-6
Degree 22
Conductor 197197
Sign −0.776−0.630i-0.776 - 0.630i
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  + (0.347 + 2.69i)2-s + (2.38 + 0.877i)3-s + (−5.19 + 1.36i)4-s + (3.26 − 1.44i)5-s + (−1.53 + 6.72i)6-s + (−2.87 − 0.371i)7-s + (−3.43 − 8.49i)8-s + (2.63 + 2.24i)9-s + (5.02 + 8.28i)10-s + (−1.40 − 2.16i)11-s + (−13.5 − 1.31i)12-s + (0.0228 + 0.355i)13-s − 7.88i·14-s + (9.05 − 0.581i)15-s + (12.3 − 6.93i)16-s + (−2.47 − 0.0795i)17-s + ⋯
L(s)  = 1  + (0.245 + 1.90i)2-s + (1.37 + 0.506i)3-s + (−2.59 + 0.681i)4-s + (1.45 − 0.645i)5-s + (−0.627 + 2.74i)6-s + (−1.08 − 0.140i)7-s + (−1.21 − 3.00i)8-s + (0.878 + 0.747i)9-s + (1.58 + 2.62i)10-s + (−0.424 − 0.651i)11-s + (−3.92 − 0.378i)12-s + (0.00633 + 0.0987i)13-s − 2.10i·14-s + (2.33 − 0.150i)15-s + (3.07 − 1.73i)16-s + (−0.601 − 0.0192i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((−0.776−0.630i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+1/2) L(s)=((−0.776−0.630i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: −0.776−0.630i-0.776 - 0.630i
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), −0.776−0.630i)(2,\ 197,\ (\ :1/2),\ -0.776 - 0.630i)

Particular Values

L(1)L(1) ≈\approx 0.609417+1.71675i0.609417 + 1.71675i
L(12)L(\frac12) ≈\approx 0.609417+1.71675i0.609417 + 1.71675i
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+(−1.05−13.9i)T 1 + (-1.05 - 13.9i)T
good2 1+(−0.347−2.69i)T+(−1.93+0.507i)T2 1 + (-0.347 - 2.69i)T + (-1.93 + 0.507i)T^{2}
3 1+(−2.38−0.877i)T+(2.28+1.94i)T2 1 + (-2.38 - 0.877i)T + (2.28 + 1.94i)T^{2}
5 1+(−3.26+1.44i)T+(3.36−3.70i)T2 1 + (-3.26 + 1.44i)T + (3.36 - 3.70i)T^{2}
7 1+(2.87+0.371i)T+(6.77+1.77i)T2 1 + (2.87 + 0.371i)T + (6.77 + 1.77i)T^{2}
11 1+(1.40+2.16i)T+(−4.45+10.0i)T2 1 + (1.40 + 2.16i)T + (-4.45 + 10.0i)T^{2}
13 1+(−0.0228−0.355i)T+(−12.8+1.66i)T2 1 + (-0.0228 - 0.355i)T + (-12.8 + 1.66i)T^{2}
17 1+(2.47+0.0795i)T+(16.9+1.08i)T2 1 + (2.47 + 0.0795i)T + (16.9 + 1.08i)T^{2}
19 1+(0.242−0.304i)T+(−4.22−18.5i)T2 1 + (0.242 - 0.304i)T + (-4.22 - 18.5i)T^{2}
23 1+(−3.39−1.12i)T+(18.4+13.7i)T2 1 + (-3.39 - 1.12i)T + (18.4 + 13.7i)T^{2}
29 1+(−0.769+7.97i)T+(−28.4−5.54i)T2 1 + (-0.769 + 7.97i)T + (-28.4 - 5.54i)T^{2}
31 1+(6.72−6.94i)T+(−0.993−30.9i)T2 1 + (6.72 - 6.94i)T + (-0.993 - 30.9i)T^{2}
37 1+(−2.04−1.15i)T+(19.1+31.6i)T2 1 + (-2.04 - 1.15i)T + (19.1 + 31.6i)T^{2}
41 1+(0.313−9.78i)T+(−40.9−2.62i)T2 1 + (0.313 - 9.78i)T + (-40.9 - 2.62i)T^{2}
43 1+(2.92−1.90i)T+(17.4−39.3i)T2 1 + (2.92 - 1.90i)T + (17.4 - 39.3i)T^{2}
47 1+(−6.31−9.04i)T+(−16.2+44.1i)T2 1 + (-6.31 - 9.04i)T + (-16.2 + 44.1i)T^{2}
53 1+(0.509+0.0992i)T+(49.1+19.8i)T2 1 + (0.509 + 0.0992i)T + (49.1 + 19.8i)T^{2}
59 1+(−1.02+1.68i)T+(−27.2−52.3i)T2 1 + (-1.02 + 1.68i)T + (-27.2 - 52.3i)T^{2}
61 1+(−1.53−4.16i)T+(−46.4+39.5i)T2 1 + (-1.53 - 4.16i)T + (-46.4 + 39.5i)T^{2}
67 1+(2.67−1.86i)T+(23.1−62.8i)T2 1 + (2.67 - 1.86i)T + (23.1 - 62.8i)T^{2}
71 1+(0.0135−0.0158i)T+(−11.3−70.0i)T2 1 + (0.0135 - 0.0158i)T + (-11.3 - 70.0i)T^{2}
73 1+(−4.63+8.22i)T+(−37.8−62.4i)T2 1 + (-4.63 + 8.22i)T + (-37.8 - 62.4i)T^{2}
79 1+(−0.520−0.230i)T+(53.1+58.4i)T2 1 + (-0.520 - 0.230i)T + (53.1 + 58.4i)T^{2}
83 1+(10.2+12.8i)T+(−18.4+80.9i)T2 1 + (10.2 + 12.8i)T + (-18.4 + 80.9i)T^{2}
89 1+(−3.22−3.33i)T+(−2.85+88.9i)T2 1 + (-3.22 - 3.33i)T + (-2.85 + 88.9i)T^{2}
97 1+(−5.67+10.8i)T+(−55.4−79.5i)T2 1 + (-5.67 + 10.8i)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

−13.23513432180635305994543469376, −12.99588938353789312344113052255, −10.09029046059736766310240093687, −9.310271476717173619437172554811, −8.919153357687933343502239905809, −7.907497808532381795424996364632, −6.56904150469092969228595619313, −5.72823404151821316729093016183, −4.52281235458261474236054826641, −3.10901679218620463295156717792, 2.00504065020792157007453849771, 2.60828385243588781137275803420, 3.60248627666923479691794508149, 5.44182765384508752156206015798, 6.99653583665974419290699337184, 8.820283495218757923127113288394, 9.369568668359902857542141005924, 10.11173156648671771418410487005, 10.91264852731586809117794863860, 12.57232003353270916567887652163

Graph of the ZZ-function along the critical line