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Properties

Label 2-684-171.7-c1-0-18
Degree 22
Conductor 684684
Sign −0.0293+0.999i-0.0293 + 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  + (1.24 − 1.20i)3-s + (0.745 − 1.29i)5-s + (0.903 − 1.56i)7-s + (0.118 − 2.99i)9-s + (0.400 − 0.694i)11-s − 1.50·13-s + (−0.618 − 2.50i)15-s + (−1.44 − 2.49i)17-s + (−2.63 + 3.47i)19-s + (−0.750 − 3.03i)21-s + 3.01·23-s + (1.38 + 2.40i)25-s + (−3.45 − 3.88i)27-s + (2.82 + 4.90i)29-s + (−1.97 − 3.42i)31-s + ⋯
L(s)  = 1  + (0.720 − 0.693i)3-s + (0.333 − 0.577i)5-s + (0.341 − 0.591i)7-s + (0.0393 − 0.999i)9-s + (0.120 − 0.209i)11-s − 0.417·13-s + (−0.159 − 0.647i)15-s + (−0.349 − 0.605i)17-s + (−0.604 + 0.796i)19-s + (−0.163 − 0.663i)21-s + 0.628·23-s + (0.277 + 0.481i)25-s + (−0.664 − 0.747i)27-s + (0.525 + 0.910i)29-s + (−0.355 − 0.615i)31-s + ⋯

Functional equation

Λ(s)=(684s/2 ΓC(s) L(s)=((−0.0293+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0293 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(684s/2 ΓC(s+1/2) L(s)=((−0.0293+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0293 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 684684    =    22⋅32⋅192^{2} \cdot 3^{2} \cdot 19
Sign: −0.0293+0.999i-0.0293 + 0.999i
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), −0.0293+0.999i)(2,\ 684,\ (\ :1/2),\ -0.0293 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.38468−1.42589i1.38468 - 1.42589i
L(12)L(\frac12) ≈\approx 1.38468−1.42589i1.38468 - 1.42589i
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+(−1.24+1.20i)T 1 + (-1.24 + 1.20i)T
19 1+(2.63−3.47i)T 1 + (2.63 - 3.47i)T
good5 1+(−0.745+1.29i)T+(−2.5−4.33i)T2 1 + (-0.745 + 1.29i)T + (-2.5 - 4.33i)T^{2}
7 1+(−0.903+1.56i)T+(−3.5−6.06i)T2 1 + (-0.903 + 1.56i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.400+0.694i)T+(−5.5−9.52i)T2 1 + (-0.400 + 0.694i)T + (-5.5 - 9.52i)T^{2}
13 1+1.50T+13T2 1 + 1.50T + 13T^{2}
17 1+(1.44+2.49i)T+(−8.5+14.7i)T2 1 + (1.44 + 2.49i)T + (-8.5 + 14.7i)T^{2}
23 1−3.01T+23T2 1 - 3.01T + 23T^{2}
29 1+(−2.82−4.90i)T+(−14.5+25.1i)T2 1 + (-2.82 - 4.90i)T + (-14.5 + 25.1i)T^{2}
31 1+(1.97+3.42i)T+(−15.5+26.8i)T2 1 + (1.97 + 3.42i)T + (-15.5 + 26.8i)T^{2}
37 1+3.08T+37T2 1 + 3.08T + 37T^{2}
41 1+(−3.66+6.34i)T+(−20.5−35.5i)T2 1 + (-3.66 + 6.34i)T + (-20.5 - 35.5i)T^{2}
43 1−3.66T+43T2 1 - 3.66T + 43T^{2}
47 1+(−0.0411−0.0713i)T+(−23.5+40.7i)T2 1 + (-0.0411 - 0.0713i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.16+2.01i)T+(−26.5−45.8i)T2 1 + (-1.16 + 2.01i)T + (-26.5 - 45.8i)T^{2}
59 1+(−3.33+5.77i)T+(−29.5−51.0i)T2 1 + (-3.33 + 5.77i)T + (-29.5 - 51.0i)T^{2}
61 1+(−5.37−9.30i)T+(−30.5+52.8i)T2 1 + (-5.37 - 9.30i)T + (-30.5 + 52.8i)T^{2}
67 1+8.50T+67T2 1 + 8.50T + 67T^{2}
71 1+(−0.0790−0.137i)T+(−35.5+61.4i)T2 1 + (-0.0790 - 0.137i)T + (-35.5 + 61.4i)T^{2}
73 1+(−1.67−2.90i)T+(−36.5+63.2i)T2 1 + (-1.67 - 2.90i)T + (-36.5 + 63.2i)T^{2}
79 1+0.856T+79T2 1 + 0.856T + 79T^{2}
83 1+(−2.39+4.14i)T+(−41.5−71.8i)T2 1 + (-2.39 + 4.14i)T + (-41.5 - 71.8i)T^{2}
89 1+(−3.35+5.81i)T+(−44.5−77.0i)T2 1 + (-3.35 + 5.81i)T + (-44.5 - 77.0i)T^{2}
97 1+3.87T+97T2 1 + 3.87T + 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

−10.16407302489341106753224674405, −9.128702202830717354350372937041, −8.653079368143620153429605663685, −7.57600007498947330436382075691, −6.97902052577699233872274818780, −5.82477860231082146533381460313, −4.68908899265001632305841826382, −3.56191742412280979346506743250, −2.24450485225542157221479626857, −1.02706791393802880121905623657, 2.12290458415548938373368175537, 2.90571939855669024191230578125, 4.23753806858574832865633441348, 5.07916681459664336239593465381, 6.26737585747560475182150461166, 7.25418327049087348554300565966, 8.347488402839598385948240022576, 8.953186451711221311668151385719, 9.838951739428297861799588222811, 10.59793847860724954152661078716

Graph of the ZZ-function along the critical line