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Properties

Label 1-197-197.196-r0-0-0
Degree 11
Conductor 197197
Sign 11
Analytic cond. 0.9148640.914864
Root an. cond. 0.9148640.914864
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(197s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 197197
Sign: 11
Analytic conductor: 0.9148640.914864
Root analytic conductor: 0.9148640.914864
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ197(196,⋅)\chi_{197} (196, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (1, 197, (0: ), 1)(1,\ 197,\ (0:\ ),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.43322899550.4332289955
L(12)L(\frac12) ≈\approx 0.43322899550.4332289955
L(1)L(1) ≈\approx 0.47500088850.4750008885
L(1)L(1) ≈\approx 0.47500088850.4750008885

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
good2 1−T 1 - T
3 1−T 1 - T
5 1−T 1 - T
7 1+T 1 + T
11 1−T 1 - T
13 1−T 1 - T
17 1−T 1 - T
19 1+T 1 + T
23 1+T 1 + T
29 1+T 1 + T
31 1−T 1 - T
37 1+T 1 + T
41 1+T 1 + T
43 1+T 1 + T
47 1+T 1 + T
53 1+T 1 + T
59 1+T 1 + T
61 1+T 1 + T
67 1−T 1 - T
71 1−T 1 - T
73 1−T 1 - T
79 1−T 1 - T
83 1+T 1 + T
89 1−T 1 - T
97 1+T 1 + T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−26.91015549878388112275243373098, −26.69179915172111168984860069084, −24.8797145776674866369859770361, −24.10096993852439686588833832324, −23.51964211783655315769254609814, −22.20091746193614388946818149208, −21.11312455363948113257821184828, −20.161410190655384488370337217570, −19.101041912396701593884952390017, −18.11625822723174927712180828683, −17.551313996657898695121121330989, −16.41218087657527046400338574975, −15.658543266406915149414498187174, −14.78558369990380964076016332804, −12.793928035823971873550369159217, −11.77103220035168417289622195039, −11.1348867056033258260813644738, −10.32794976444752957642728447190, −8.90405576737827996770584228794, −7.60149056272632292052777748313, −7.18845902309268000133166862156, −5.51331778593733928444594361840, −4.47513802839947470737032323861, −2.53623353923091238315535545203, −0.82624711791797162386794860521, 0.82624711791797162386794860521, 2.53623353923091238315535545203, 4.47513802839947470737032323861, 5.51331778593733928444594361840, 7.18845902309268000133166862156, 7.60149056272632292052777748313, 8.90405576737827996770584228794, 10.32794976444752957642728447190, 11.1348867056033258260813644738, 11.77103220035168417289622195039, 12.793928035823971873550369159217, 14.78558369990380964076016332804, 15.658543266406915149414498187174, 16.41218087657527046400338574975, 17.551313996657898695121121330989, 18.11625822723174927712180828683, 19.101041912396701593884952390017, 20.161410190655384488370337217570, 21.11312455363948113257821184828, 22.20091746193614388946818149208, 23.51964211783655315769254609814, 24.10096993852439686588833832324, 24.8797145776674866369859770361, 26.69179915172111168984860069084, 26.91015549878388112275243373098

Graph of the ZZ-function along the critical line