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Properties

Label 2-684-171.7-c1-0-16
Degree 22
Conductor 684684
Sign 0.887+0.460i0.887 + 0.460i
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  + (1.23 + 1.21i)3-s + (2.12 − 3.67i)5-s + (1.38 − 2.39i)7-s + (0.0360 + 2.99i)9-s + (−1.31 + 2.26i)11-s − 0.0946·13-s + (7.09 − 1.94i)15-s + (−0.170 − 0.295i)17-s + (−1.77 − 3.97i)19-s + (4.61 − 1.26i)21-s + 4.62·23-s + (−6.51 − 11.2i)25-s + (−3.60 + 3.73i)27-s + (0.706 + 1.22i)29-s + (1.43 + 2.48i)31-s + ⋯
L(s)  = 1  + (0.711 + 0.702i)3-s + (0.949 − 1.64i)5-s + (0.522 − 0.904i)7-s + (0.0120 + 0.999i)9-s + (−0.395 + 0.684i)11-s − 0.0262·13-s + (1.83 − 0.502i)15-s + (−0.0413 − 0.0715i)17-s + (−0.408 − 0.912i)19-s + (1.00 − 0.276i)21-s + 0.963·23-s + (−1.30 − 2.25i)25-s + (−0.694 + 0.719i)27-s + (0.131 + 0.227i)29-s + (0.257 + 0.446i)31-s + ⋯

Functional equation

Λ(s)=(684s/2 ΓC(s) L(s)=((0.887+0.460i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(684s/2 ΓC(s+1/2) L(s)=((0.887+0.460i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 684684    =    22⋅32⋅192^{2} \cdot 3^{2} \cdot 19
Sign: 0.887+0.460i0.887 + 0.460i
Analytic conductor: 5.461765.46176
Root analytic conductor: 2.337042.33704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ684(349,⋅)\chi_{684} (349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 684, ( :1/2), 0.887+0.460i)(2,\ 684,\ (\ :1/2),\ 0.887 + 0.460i)

Particular Values

L(1)L(1) ≈\approx 2.20019−0.536865i2.20019 - 0.536865i
L(12)L(\frac12) ≈\approx 2.20019−0.536865i2.20019 - 0.536865i
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+(−1.23−1.21i)T 1 + (-1.23 - 1.21i)T
19 1+(1.77+3.97i)T 1 + (1.77 + 3.97i)T
good5 1+(−2.12+3.67i)T+(−2.5−4.33i)T2 1 + (-2.12 + 3.67i)T + (-2.5 - 4.33i)T^{2}
7 1+(−1.38+2.39i)T+(−3.5−6.06i)T2 1 + (-1.38 + 2.39i)T + (-3.5 - 6.06i)T^{2}
11 1+(1.31−2.26i)T+(−5.5−9.52i)T2 1 + (1.31 - 2.26i)T + (-5.5 - 9.52i)T^{2}
13 1+0.0946T+13T2 1 + 0.0946T + 13T^{2}
17 1+(0.170+0.295i)T+(−8.5+14.7i)T2 1 + (0.170 + 0.295i)T + (-8.5 + 14.7i)T^{2}
23 1−4.62T+23T2 1 - 4.62T + 23T^{2}
29 1+(−0.706−1.22i)T+(−14.5+25.1i)T2 1 + (-0.706 - 1.22i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.43−2.48i)T+(−15.5+26.8i)T2 1 + (-1.43 - 2.48i)T + (-15.5 + 26.8i)T^{2}
37 1+10.6T+37T2 1 + 10.6T + 37T^{2}
41 1+(−1.21+2.10i)T+(−20.5−35.5i)T2 1 + (-1.21 + 2.10i)T + (-20.5 - 35.5i)T^{2}
43 1−12.5T+43T2 1 - 12.5T + 43T^{2}
47 1+(−3.42−5.92i)T+(−23.5+40.7i)T2 1 + (-3.42 - 5.92i)T + (-23.5 + 40.7i)T^{2}
53 1+(2.21−3.83i)T+(−26.5−45.8i)T2 1 + (2.21 - 3.83i)T + (-26.5 - 45.8i)T^{2}
59 1+(6.77−11.7i)T+(−29.5−51.0i)T2 1 + (6.77 - 11.7i)T + (-29.5 - 51.0i)T^{2}
61 1+(0.658+1.14i)T+(−30.5+52.8i)T2 1 + (0.658 + 1.14i)T + (-30.5 + 52.8i)T^{2}
67 1−7.22T+67T2 1 - 7.22T + 67T^{2}
71 1+(−5.00−8.67i)T+(−35.5+61.4i)T2 1 + (-5.00 - 8.67i)T + (-35.5 + 61.4i)T^{2}
73 1+(4.76+8.24i)T+(−36.5+63.2i)T2 1 + (4.76 + 8.24i)T + (-36.5 + 63.2i)T^{2}
79 1−3.53T+79T2 1 - 3.53T + 79T^{2}
83 1+(1.93−3.35i)T+(−41.5−71.8i)T2 1 + (1.93 - 3.35i)T + (-41.5 - 71.8i)T^{2}
89 1+(5.91−10.2i)T+(−44.5−77.0i)T2 1 + (5.91 - 10.2i)T + (-44.5 - 77.0i)T^{2}
97 1+0.430T+97T2 1 + 0.430T + 97T^{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

−10.35164761089646785721156455173, −9.302048680747654237510244464632, −8.968502905001915389421720118088, −8.032860135590310545191301847535, −7.09359151136926624274195915735, −5.50596219319524333426845197137, −4.76463336532856758691372249533, −4.23336719932705701038536569466, −2.51534515430768864104264198256, −1.27226117347128732994584588463, 1.88819157180054359644833073166, 2.64139568848388728836208206072, 3.52509941636821749643471538615, 5.48575187158122205426345982473, 6.19069173796888424393635136454, 6.98759318890235949615112636826, 7.916379252272217415917307662626, 8.768474653498457231152290650075, 9.643759616142362749571224689850, 10.57867826427003021579266021326

Graph of the ZZ-function along the critical line