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Properties

Label 2-197-197.100-c1-0-7
Degree 22
Conductor 197197
Sign −0.340+0.940i-0.340 + 0.940i
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  + (−2.71 + 0.710i)2-s + (−1.73 − 1.47i)3-s + (5.09 − 2.87i)4-s + (2.16 − 2.38i)5-s + (5.74 + 2.76i)6-s + (−0.311 − 0.0815i)7-s + (−7.75 + 7.51i)8-s + (0.347 + 2.14i)9-s + (−4.17 + 8.00i)10-s + (0.579 − 1.30i)11-s + (−13.0 − 2.54i)12-s + (5.35 − 0.690i)13-s + 0.901·14-s + (−7.26 + 0.937i)15-s + (9.61 − 15.8i)16-s + (−1.74 − 0.112i)17-s + ⋯
L(s)  = 1  + (−1.91 + 0.502i)2-s + (−1.00 − 0.851i)3-s + (2.54 − 1.43i)4-s + (0.968 − 1.06i)5-s + (2.34 + 1.12i)6-s + (−0.117 − 0.0308i)7-s + (−2.74 + 2.65i)8-s + (0.115 + 0.715i)9-s + (−1.32 + 2.53i)10-s + (0.174 − 0.394i)11-s + (−3.77 − 0.734i)12-s + (1.48 − 0.191i)13-s + 0.240·14-s + (−1.87 + 0.241i)15-s + (2.40 − 3.96i)16-s + (−0.423 − 0.0272i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((−0.340+0.940i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+1/2) L(s)=((−0.340+0.940i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: −0.340+0.940i-0.340 + 0.940i
Analytic conductor: 1.573051.57305
Root analytic conductor: 1.254211.25421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ197(100,⋅)\chi_{197} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :1/2), −0.340+0.940i)(2,\ 197,\ (\ :1/2),\ -0.340 + 0.940i)

Particular Values

L(1)L(1) ≈\approx 0.227809−0.324889i0.227809 - 0.324889i
L(12)L(\frac12) ≈\approx 0.227809−0.324889i0.227809 - 0.324889i
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+(3.06+13.6i)T 1 + (3.06 + 13.6i)T
good2 1+(2.71−0.710i)T+(1.74−0.981i)T2 1 + (2.71 - 0.710i)T + (1.74 - 0.981i)T^{2}
3 1+(1.73+1.47i)T+(0.478+2.96i)T2 1 + (1.73 + 1.47i)T + (0.478 + 2.96i)T^{2}
5 1+(−2.16+2.38i)T+(−0.480−4.97i)T2 1 + (-2.16 + 2.38i)T + (-0.480 - 4.97i)T^{2}
7 1+(0.311+0.0815i)T+(6.09+3.43i)T2 1 + (0.311 + 0.0815i)T + (6.09 + 3.43i)T^{2}
11 1+(−0.579+1.30i)T+(−7.39−8.14i)T2 1 + (-0.579 + 1.30i)T + (-7.39 - 8.14i)T^{2}
13 1+(−5.35+0.690i)T+(12.5−3.29i)T2 1 + (-5.35 + 0.690i)T + (12.5 - 3.29i)T^{2}
17 1+(1.74+0.112i)T+(16.8+2.17i)T2 1 + (1.74 + 0.112i)T + (16.8 + 2.17i)T^{2}
19 1+(0.225+0.989i)T+(−17.1+8.24i)T2 1 + (0.225 + 0.989i)T + (-17.1 + 8.24i)T^{2}
23 1+(5.87+4.38i)T+(6.54+22.0i)T2 1 + (5.87 + 4.38i)T + (6.54 + 22.0i)T^{2}
29 1+(0.371+0.0724i)T+(26.8+10.8i)T2 1 + (0.371 + 0.0724i)T + (26.8 + 10.8i)T^{2}
31 1+(−0.0695−2.16i)T+(−30.9+1.98i)T2 1 + (-0.0695 - 2.16i)T + (-30.9 + 1.98i)T^{2}
37 1+(3.78+6.25i)T+(−17.1+32.8i)T2 1 + (3.78 + 6.25i)T + (-17.1 + 32.8i)T^{2}
41 1+(−0.272−0.0175i)T+(40.6+5.24i)T2 1 + (-0.272 - 0.0175i)T + (40.6 + 5.24i)T^{2}
43 1+(−0.676+1.52i)T+(−28.9−31.8i)T2 1 + (-0.676 + 1.52i)T + (-28.9 - 31.8i)T^{2}
47 1+(−1.05+2.87i)T+(−35.7−30.4i)T2 1 + (-1.05 + 2.87i)T + (-35.7 - 30.4i)T^{2}
53 1+(−6.87−2.78i)T+(38.0+36.8i)T2 1 + (-6.87 - 2.78i)T + (38.0 + 36.8i)T^{2}
59 1+(−2.32−4.45i)T+(−33.7+48.3i)T2 1 + (-2.32 - 4.45i)T + (-33.7 + 48.3i)T^{2}
61 1+(6.39−5.44i)T+(9.73−60.2i)T2 1 + (6.39 - 5.44i)T + (9.73 - 60.2i)T^{2}
67 1+(1.71−4.66i)T+(−51.0−43.4i)T2 1 + (1.71 - 4.66i)T + (-51.0 - 43.4i)T^{2}
71 1+(−0.809−5.00i)T+(−67.3+22.3i)T2 1 + (-0.809 - 5.00i)T + (-67.3 + 22.3i)T^{2}
73 1+(−8.45−13.9i)T+(−33.7+64.7i)T2 1 + (-8.45 - 13.9i)T + (-33.7 + 64.7i)T^{2}
79 1+(−0.481−0.529i)T+(−7.58+78.6i)T2 1 + (-0.481 - 0.529i)T + (-7.58 + 78.6i)T^{2}
83 1+(−2.02+8.88i)T+(−74.7−36.0i)T2 1 + (-2.02 + 8.88i)T + (-74.7 - 36.0i)T^{2}
89 1+(−0.342+10.6i)T+(−88.8−5.70i)T2 1 + (-0.342 + 10.6i)T + (-88.8 - 5.70i)T^{2}
97 1+(3.35+4.80i)T+(−33.5+91.0i)T2 1 + (3.35 + 4.80i)T + (-33.5 + 91.0i)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

−11.87151523130749699038697236901, −10.98216264405659207499967031437, −10.11489064639826316458440933046, −8.919607772101589708647370985987, −8.453210618130742770406641817602, −7.04262115440058417756307760675, −6.09469482082494236542572486488, −5.65328539812188520506504431308, −1.82725357673576165911590519369, −0.73006402595193429844391398028, 1.90757575618628590863910621158, 3.55635110508069162038798843591, 6.05586324242357819824319640133, 6.55516811379251778811522973005, 7.982025400824696919687702729286, 9.325353050739661844511591683665, 9.936371344769342376492203252148, 10.69553992020443522911366543092, 11.17316901832508278236770964585, 12.06272821000927038082103277364

Graph of the ZZ-function along the critical line