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Properties

Label 2-693-1.1-c3-0-25
Degree 22
Conductor 693693
Sign 11
Analytic cond. 40.888340.8883
Root an. cond. 6.394396.39439
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.45·2-s + 11.8·4-s + 12.1·5-s + 7·7-s − 17.2·8-s − 54.3·10-s + 11·11-s − 15.6·13-s − 31.1·14-s − 18.1·16-s + 43.3·17-s + 51.7·19-s + 144.·20-s − 49.0·22-s + 121.·23-s + 23.6·25-s + 69.7·26-s + 83.0·28-s + 187.·29-s + 91.9·31-s + 218.·32-s − 193.·34-s + 85.3·35-s − 226.·37-s − 230.·38-s − 209.·40-s − 11.2·41-s + ⋯
L(s)  = 1  − 1.57·2-s + 1.48·4-s + 1.09·5-s + 0.377·7-s − 0.760·8-s − 1.71·10-s + 0.301·11-s − 0.333·13-s − 0.595·14-s − 0.283·16-s + 0.618·17-s + 0.625·19-s + 1.61·20-s − 0.475·22-s + 1.09·23-s + 0.188·25-s + 0.526·26-s + 0.560·28-s + 1.19·29-s + 0.532·31-s + 1.20·32-s − 0.973·34-s + 0.412·35-s − 1.00·37-s − 0.985·38-s − 0.829·40-s − 0.0430·41-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 40.888340.8883
Root analytic conductor: 6.394396.39439
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 693, ( :3/2), 1)(2,\ 693,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.2911781891.291178189
L(12)L(\frac12) ≈\approx 1.2911781891.291178189
L(52)L(\frac{5}{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)
bad3 1 1
7 1−7T 1 - 7T
11 1−11T 1 - 11T
good2 1+4.45T+8T2 1 + 4.45T + 8T^{2}
5 1−12.1T+125T2 1 - 12.1T + 125T^{2}
13 1+15.6T+2.19e3T2 1 + 15.6T + 2.19e3T^{2}
17 1−43.3T+4.91e3T2 1 - 43.3T + 4.91e3T^{2}
19 1−51.7T+6.85e3T2 1 - 51.7T + 6.85e3T^{2}
23 1−121.T+1.21e4T2 1 - 121.T + 1.21e4T^{2}
29 1−187.T+2.43e4T2 1 - 187.T + 2.43e4T^{2}
31 1−91.9T+2.97e4T2 1 - 91.9T + 2.97e4T^{2}
37 1+226.T+5.06e4T2 1 + 226.T + 5.06e4T^{2}
41 1+11.2T+6.89e4T2 1 + 11.2T + 6.89e4T^{2}
43 1+98.0T+7.95e4T2 1 + 98.0T + 7.95e4T^{2}
47 1−186.T+1.03e5T2 1 - 186.T + 1.03e5T^{2}
53 1−487.T+1.48e5T2 1 - 487.T + 1.48e5T^{2}
59 1+697.T+2.05e5T2 1 + 697.T + 2.05e5T^{2}
61 1−486.T+2.26e5T2 1 - 486.T + 2.26e5T^{2}
67 1+436.T+3.00e5T2 1 + 436.T + 3.00e5T^{2}
71 1+715.T+3.57e5T2 1 + 715.T + 3.57e5T^{2}
73 1+860.T+3.89e5T2 1 + 860.T + 3.89e5T^{2}
79 1−157.T+4.93e5T2 1 - 157.T + 4.93e5T^{2}
83 1−397.T+5.71e5T2 1 - 397.T + 5.71e5T^{2}
89 1+237.T+7.04e5T2 1 + 237.T + 7.04e5T^{2}
97 1−590.T+9.12e5T2 1 - 590.T + 9.12e5T^{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

−10.07428954119634228263837995444, −9.173638662485378044106422717431, −8.614490607159946114126066581415, −7.58125019211478982373393901256, −6.84555090430115715223854636795, −5.80969932318821890562048343018, −4.74430683349647594119378387895, −2.92603031935082200278204693207, −1.77843542542002214978940912404, −0.884472823845140951219432637219, 0.884472823845140951219432637219, 1.77843542542002214978940912404, 2.92603031935082200278204693207, 4.74430683349647594119378387895, 5.80969932318821890562048343018, 6.84555090430115715223854636795, 7.58125019211478982373393901256, 8.614490607159946114126066581415, 9.173638662485378044106422717431, 10.07428954119634228263837995444

Graph of the ZZ-function along the critical line