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Properties

Label 2-197-197.14-c2-0-10
Degree 22
Conductor 197197
Sign −0.658−0.752i-0.658 - 0.752i
Analytic cond. 5.367865.36786
Root an. cond. 2.316862.31686
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.11 + 1.11i)2-s + (2.19 − 2.19i)3-s + 1.52i·4-s + (−1.87 + 1.87i)5-s + 4.88i·6-s + 5.07i·7-s + (−6.14 − 6.14i)8-s − 0.630i·9-s − 4.17i·10-s + (−9.24 + 9.24i)11-s + (3.33 + 3.33i)12-s + (2.79 − 2.79i)13-s + (−5.64 − 5.64i)14-s + 8.22i·15-s + 7.60·16-s + (−12.5 − 12.5i)17-s + ⋯
L(s)  = 1  + (−0.556 + 0.556i)2-s + (0.731 − 0.731i)3-s + 0.380i·4-s + (−0.374 + 0.374i)5-s + 0.814i·6-s + 0.724i·7-s + (−0.768 − 0.768i)8-s − 0.0701i·9-s − 0.417i·10-s + (−0.840 + 0.840i)11-s + (0.277 + 0.277i)12-s + (0.214 − 0.214i)13-s + (−0.403 − 0.403i)14-s + 0.548i·15-s + 0.475·16-s + (−0.736 − 0.736i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((−0.658−0.752i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+1) L(s)=((−0.658−0.752i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: −0.658−0.752i-0.658 - 0.752i
Analytic conductor: 5.367865.36786
Root analytic conductor: 2.316862.31686
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ197(14,⋅)\chi_{197} (14, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :1), −0.658−0.752i)(2,\ 197,\ (\ :1),\ -0.658 - 0.752i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.400594+0.882946i0.400594 + 0.882946i
L(12)L(\frac12) ≈\approx 0.400594+0.882946i0.400594 + 0.882946i
L(2)L(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+(6.06−196.i)T 1 + (6.06 - 196. i)T
good2 1+(1.11−1.11i)T−4iT2 1 + (1.11 - 1.11i)T - 4iT^{2}
3 1+(−2.19+2.19i)T−9iT2 1 + (-2.19 + 2.19i)T - 9iT^{2}
5 1+(1.87−1.87i)T−25iT2 1 + (1.87 - 1.87i)T - 25iT^{2}
7 1−5.07iT−49T2 1 - 5.07iT - 49T^{2}
11 1+(9.24−9.24i)T−121iT2 1 + (9.24 - 9.24i)T - 121iT^{2}
13 1+(−2.79+2.79i)T−169iT2 1 + (-2.79 + 2.79i)T - 169iT^{2}
17 1+(12.5+12.5i)T+289iT2 1 + (12.5 + 12.5i)T + 289iT^{2}
19 1−33.2iT−361T2 1 - 33.2iT - 361T^{2}
23 1−30.3T+529T2 1 - 30.3T + 529T^{2}
29 1+36.4T+841T2 1 + 36.4T + 841T^{2}
31 1+(5.16−5.16i)T−961iT2 1 + (5.16 - 5.16i)T - 961iT^{2}
37 1+42.9T+1.36e3T2 1 + 42.9T + 1.36e3T^{2}
41 1−0.230iT−1.68e3T2 1 - 0.230iT - 1.68e3T^{2}
43 1+70.5iT−1.84e3T2 1 + 70.5iT - 1.84e3T^{2}
47 1−55.0iT−2.20e3T2 1 - 55.0iT - 2.20e3T^{2}
53 1−74.5T+2.80e3T2 1 - 74.5T + 2.80e3T^{2}
59 1−75.7T+3.48e3T2 1 - 75.7T + 3.48e3T^{2}
61 1−83.6T+3.72e3T2 1 - 83.6T + 3.72e3T^{2}
67 1+(16.2−16.2i)T−4.48e3iT2 1 + (16.2 - 16.2i)T - 4.48e3iT^{2}
71 1+(65.9+65.9i)T+5.04e3iT2 1 + (65.9 + 65.9i)T + 5.04e3iT^{2}
73 1+(−53.0+53.0i)T−5.32e3iT2 1 + (-53.0 + 53.0i)T - 5.32e3iT^{2}
79 1+(−39.8−39.8i)T+6.24e3iT2 1 + (-39.8 - 39.8i)T + 6.24e3iT^{2}
83 1+121.iT−6.88e3T2 1 + 121. iT - 6.88e3T^{2}
89 1+(81.1+81.1i)T+7.92e3iT2 1 + (81.1 + 81.1i)T + 7.92e3iT^{2}
97 1−168.iT−9.40e3T2 1 - 168. iT - 9.40e3T^{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

−12.73226401051962115661531045156, −11.86585402531543586525079782322, −10.52817686789098658184311763606, −9.187533702111006800091228675610, −8.449930388144303976735236468298, −7.46855231660282221457369602358, −7.07376441517262045430074110141, −5.42952585662251116690812500243, −3.47494695542059710337963504691, −2.24272951333048672610759400019, 0.59817383931822810092612788912, 2.66169053047255502330082588743, 3.99206395103038069077303644419, 5.25196946762335029852093061605, 6.85496984061652156723382527871, 8.489156494175677675596142870333, 8.869271111878537572164542681326, 9.939666037255036639538501247063, 10.84403394388024860374109907101, 11.39377607880859985183057283987

Graph of the ZZ-function along the critical line