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Properties

Label 2-684-171.106-c1-0-9
Degree 22
Conductor 684684
Sign 0.0408+0.999i0.0408 + 0.999i
Analytic cond. 5.461765.46176
Root an. cond. 2.337042.33704
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  + (−1.04 + 1.38i)3-s − 2.40·5-s + (−1.53 + 2.65i)7-s + (−0.810 − 2.88i)9-s + (−2.96 + 5.14i)11-s + (2.81 − 4.88i)13-s + (2.51 − 3.32i)15-s + (1.15 − 2.00i)17-s + (1.16 − 4.19i)19-s + (−2.05 − 4.89i)21-s + (−0.654 + 1.13i)23-s + 0.787·25-s + (4.83 + 1.90i)27-s + 6.77·29-s + (−4.86 − 8.42i)31-s + ⋯
L(s)  = 1  + (−0.604 + 0.796i)3-s − 1.07·5-s + (−0.578 + 1.00i)7-s + (−0.270 − 0.962i)9-s + (−0.894 + 1.55i)11-s + (0.782 − 1.35i)13-s + (0.649 − 0.857i)15-s + (0.280 − 0.485i)17-s + (0.267 − 0.963i)19-s + (−0.449 − 1.06i)21-s + (−0.136 + 0.236i)23-s + 0.157·25-s + (0.930 + 0.366i)27-s + 1.25·29-s + (−0.873 − 1.51i)31-s + ⋯

Functional equation

Λ(s)=(684s/2 ΓC(s) L(s)=((0.0408+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0408 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(684s/2 ΓC(s+1/2) L(s)=((0.0408+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0408 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 684684    =    22⋅32⋅192^{2} \cdot 3^{2} \cdot 19
Sign: 0.0408+0.999i0.0408 + 0.999i
Analytic conductor: 5.461765.46176
Root analytic conductor: 2.337042.33704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ684(277,⋅)\chi_{684} (277, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 684, ( :1/2), 0.0408+0.999i)(2,\ 684,\ (\ :1/2),\ 0.0408 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.172339−0.165436i0.172339 - 0.165436i
L(12)L(\frac12) ≈\approx 0.172339−0.165436i0.172339 - 0.165436i
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)
bad2 1 1
3 1+(1.04−1.38i)T 1 + (1.04 - 1.38i)T
19 1+(−1.16+4.19i)T 1 + (-1.16 + 4.19i)T
good5 1+2.40T+5T2 1 + 2.40T + 5T^{2}
7 1+(1.53−2.65i)T+(−3.5−6.06i)T2 1 + (1.53 - 2.65i)T + (-3.5 - 6.06i)T^{2}
11 1+(2.96−5.14i)T+(−5.5−9.52i)T2 1 + (2.96 - 5.14i)T + (-5.5 - 9.52i)T^{2}
13 1+(−2.81+4.88i)T+(−6.5−11.2i)T2 1 + (-2.81 + 4.88i)T + (-6.5 - 11.2i)T^{2}
17 1+(−1.15+2.00i)T+(−8.5−14.7i)T2 1 + (-1.15 + 2.00i)T + (-8.5 - 14.7i)T^{2}
23 1+(0.654−1.13i)T+(−11.5−19.9i)T2 1 + (0.654 - 1.13i)T + (-11.5 - 19.9i)T^{2}
29 1−6.77T+29T2 1 - 6.77T + 29T^{2}
31 1+(4.86+8.42i)T+(−15.5+26.8i)T2 1 + (4.86 + 8.42i)T + (-15.5 + 26.8i)T^{2}
37 1+2.91T+37T2 1 + 2.91T + 37T^{2}
41 1+3.91T+41T2 1 + 3.91T + 41T^{2}
43 1+(0.0602+0.104i)T+(−21.5+37.2i)T2 1 + (0.0602 + 0.104i)T + (-21.5 + 37.2i)T^{2}
47 1+10.0T+47T2 1 + 10.0T + 47T^{2}
53 1+(4.85+8.41i)T+(−26.5+45.8i)T2 1 + (4.85 + 8.41i)T + (-26.5 + 45.8i)T^{2}
59 1−8.36T+59T2 1 - 8.36T + 59T^{2}
61 1−9.94T+61T2 1 - 9.94T + 61T^{2}
67 1+(−5.03+8.72i)T+(−33.5−58.0i)T2 1 + (-5.03 + 8.72i)T + (-33.5 - 58.0i)T^{2}
71 1+(1.00−1.73i)T+(−35.5−61.4i)T2 1 + (1.00 - 1.73i)T + (-35.5 - 61.4i)T^{2}
73 1+(4.09−7.10i)T+(−36.5−63.2i)T2 1 + (4.09 - 7.10i)T + (-36.5 - 63.2i)T^{2}
79 1+(5.70+9.88i)T+(−39.5+68.4i)T2 1 + (5.70 + 9.88i)T + (-39.5 + 68.4i)T^{2}
83 1+(2.77−4.79i)T+(−41.5−71.8i)T2 1 + (2.77 - 4.79i)T + (-41.5 - 71.8i)T^{2}
89 1+(4.48+7.76i)T+(−44.5+77.0i)T2 1 + (4.48 + 7.76i)T + (-44.5 + 77.0i)T^{2}
97 1+(3.43+5.94i)T+(−48.5+84.0i)T2 1 + (3.43 + 5.94i)T + (-48.5 + 84.0i)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

−10.14626090537871023743015127011, −9.667694441241046823229402917963, −8.560322336303508901088171856978, −7.71492172743610996736249665567, −6.67563433049463548615798224008, −5.50216074994540392922269653028, −4.89211453342603558962411754844, −3.70318661592580024131849984954, −2.73275068356261620783521161849, −0.15036194332776219697060408211, 1.26216060605804121714939882419, 3.24076092912795527119664760624, 4.06007225616521542944856898926, 5.41459274234205605577551542564, 6.44532378597224373293675327304, 7.07120807693670211844046121351, 8.135555735470160378796431540586, 8.511593728947381032890867110032, 10.17401362077193193015993888724, 10.84979824646347076765435118927

Graph of the ZZ-function along the critical line