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Properties

Label 2-684-19.15-c0-0-0
Degree 22
Conductor 684684
Sign 0.992+0.120i0.992 + 0.120i
Analytic cond. 0.3413600.341360
Root an. cond. 0.5842600.584260
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.300i)7-s + (1.26 + 0.223i)13-s + (0.5 − 0.866i)19-s + (−0.173 + 0.984i)25-s + (−0.592 − 0.342i)31-s − 0.684i·37-s + (−1.43 + 1.20i)43-s + (0.439 + 0.761i)49-s + (−1.17 − 0.984i)61-s + (−0.673 − 1.85i)67-s + (0.266 + 1.50i)73-s + (−1.93 + 0.342i)79-s + (0.286 − 0.342i)91-s + (−0.592 + 1.62i)97-s + (−1.11 + 0.642i)103-s + ⋯
L(s)  = 1  + (0.173 − 0.300i)7-s + (1.26 + 0.223i)13-s + (0.5 − 0.866i)19-s + (−0.173 + 0.984i)25-s + (−0.592 − 0.342i)31-s − 0.684i·37-s + (−1.43 + 1.20i)43-s + (0.439 + 0.761i)49-s + (−1.17 − 0.984i)61-s + (−0.673 − 1.85i)67-s + (0.266 + 1.50i)73-s + (−1.93 + 0.342i)79-s + (0.286 − 0.342i)91-s + (−0.592 + 1.62i)97-s + (−1.11 + 0.642i)103-s + ⋯

Functional equation

Λ(s)=(684s/2 ΓC(s) L(s)=((0.992+0.120i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(684s/2 ΓC(s) L(s)=((0.992+0.120i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 684684    =    22⋅32⋅192^{2} \cdot 3^{2} \cdot 19
Sign: 0.992+0.120i0.992 + 0.120i
Analytic conductor: 0.3413600.341360
Root analytic conductor: 0.5842600.584260
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ684(433,⋅)\chi_{684} (433, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 684, ( :0), 0.992+0.120i)(2,\ 684,\ (\ :0),\ 0.992 + 0.120i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.99347364950.9934736495
L(12)L(\frac12) ≈\approx 0.99347364950.9934736495
L(1)L(1) 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)
bad2 1 1
3 1 1
19 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good5 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
7 1+(−0.173+0.300i)T+(−0.5−0.866i)T2 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2}
11 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
13 1+(−1.26−0.223i)T+(0.939+0.342i)T2 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2}
17 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
23 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
29 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
31 1+(0.592+0.342i)T+(0.5+0.866i)T2 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2}
37 1+0.684iT−T2 1 + 0.684iT - T^{2}
41 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
43 1+(1.43−1.20i)T+(0.173−0.984i)T2 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2}
47 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
53 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
59 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
61 1+(1.17+0.984i)T+(0.173+0.984i)T2 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2}
67 1+(0.673+1.85i)T+(−0.766+0.642i)T2 1 + (0.673 + 1.85i)T + (-0.766 + 0.642i)T^{2}
71 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
73 1+(−0.266−1.50i)T+(−0.939+0.342i)T2 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2}
79 1+(1.93−0.342i)T+(0.939−0.342i)T2 1 + (1.93 - 0.342i)T + (0.939 - 0.342i)T^{2}
83 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
89 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
97 1+(0.592−1.62i)T+(−0.766−0.642i)T2 1 + (0.592 - 1.62i)T + (-0.766 - 0.642i)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

−10.99292849380964429654084901097, −9.704030326828789823234205827512, −9.020117991550762485058564663836, −8.067705491979884392212325146904, −7.18468926952169906891068501084, −6.24709248013325533580793953248, −5.24878685478937538249821368770, −4.13736558675381106353360772261, −3.11232544877485692927436509502, −1.49219031819791772022830470178, 1.60769552276505894918811906902, 3.12840136531938841681373630438, 4.15150028260180327291670876234, 5.41229490700873079074749976410, 6.15445842382711140112471029810, 7.21501617772971098462621695038, 8.295767936786091469123624481403, 8.789692427325710914689404402639, 9.987108360232228153659880026543, 10.60826526278565807621625997636

Graph of the ZZ-function along the critical line