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Properties

Label 2-693-11.4-c1-0-14
Degree 22
Conductor 693693
Sign 0.610+0.792i0.610 + 0.792i
Analytic cond. 5.533635.53363
Root an. cond. 2.352362.35236
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  + (−0.614 − 0.446i)2-s + (−0.439 − 1.35i)4-s + (2.31 − 1.68i)5-s + (0.309 + 0.951i)7-s + (−0.803 + 2.47i)8-s − 2.17·10-s + (3.15 + 1.01i)11-s + (5.63 + 4.09i)13-s + (0.234 − 0.722i)14-s + (−0.706 + 0.512i)16-s + (5.38 − 3.91i)17-s + (−1.77 + 5.46i)19-s + (−3.29 − 2.39i)20-s + (−1.48 − 2.03i)22-s + 0.724·23-s + ⋯
L(s)  = 1  + (−0.434 − 0.315i)2-s + (−0.219 − 0.676i)4-s + (1.03 − 0.753i)5-s + (0.116 + 0.359i)7-s + (−0.284 + 0.874i)8-s − 0.688·10-s + (0.952 + 0.304i)11-s + (1.56 + 1.13i)13-s + (0.0627 − 0.193i)14-s + (−0.176 + 0.128i)16-s + (1.30 − 0.949i)17-s + (−0.407 + 1.25i)19-s + (−0.737 − 0.536i)20-s + (−0.317 − 0.432i)22-s + 0.151·23-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((0.610+0.792i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=((0.610+0.792i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 0.610+0.792i0.610 + 0.792i
Analytic conductor: 5.533635.53363
Root analytic conductor: 2.352362.35236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ693(631,⋅)\chi_{693} (631, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :1/2), 0.610+0.792i)(2,\ 693,\ (\ :1/2),\ 0.610 + 0.792i)

Particular Values

L(1)L(1) ≈\approx 1.37010−0.673864i1.37010 - 0.673864i
L(12)L(\frac12) ≈\approx 1.37010−0.673864i1.37010 - 0.673864i
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)
bad3 1 1
7 1+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
11 1+(−3.15−1.01i)T 1 + (-3.15 - 1.01i)T
good2 1+(0.614+0.446i)T+(0.618+1.90i)T2 1 + (0.614 + 0.446i)T + (0.618 + 1.90i)T^{2}
5 1+(−2.31+1.68i)T+(1.54−4.75i)T2 1 + (-2.31 + 1.68i)T + (1.54 - 4.75i)T^{2}
13 1+(−5.63−4.09i)T+(4.01+12.3i)T2 1 + (-5.63 - 4.09i)T + (4.01 + 12.3i)T^{2}
17 1+(−5.38+3.91i)T+(5.25−16.1i)T2 1 + (-5.38 + 3.91i)T + (5.25 - 16.1i)T^{2}
19 1+(1.77−5.46i)T+(−15.3−11.1i)T2 1 + (1.77 - 5.46i)T + (-15.3 - 11.1i)T^{2}
23 1−0.724T+23T2 1 - 0.724T + 23T^{2}
29 1+(2.33+7.19i)T+(−23.4+17.0i)T2 1 + (2.33 + 7.19i)T + (-23.4 + 17.0i)T^{2}
31 1+(2.28+1.65i)T+(9.57+29.4i)T2 1 + (2.28 + 1.65i)T + (9.57 + 29.4i)T^{2}
37 1+(0.532+1.63i)T+(−29.9+21.7i)T2 1 + (0.532 + 1.63i)T + (-29.9 + 21.7i)T^{2}
41 1+(0.346−1.06i)T+(−33.1−24.0i)T2 1 + (0.346 - 1.06i)T + (-33.1 - 24.0i)T^{2}
43 1+11.7T+43T2 1 + 11.7T + 43T^{2}
47 1+(1.60−4.93i)T+(−38.0−27.6i)T2 1 + (1.60 - 4.93i)T + (-38.0 - 27.6i)T^{2}
53 1+(5.06+3.67i)T+(16.3+50.4i)T2 1 + (5.06 + 3.67i)T + (16.3 + 50.4i)T^{2}
59 1+(1.23+3.78i)T+(−47.7+34.6i)T2 1 + (1.23 + 3.78i)T + (-47.7 + 34.6i)T^{2}
61 1+(−4.83+3.50i)T+(18.8−58.0i)T2 1 + (-4.83 + 3.50i)T + (18.8 - 58.0i)T^{2}
67 1−10.3T+67T2 1 - 10.3T + 67T^{2}
71 1+(0.805−0.584i)T+(21.9−67.5i)T2 1 + (0.805 - 0.584i)T + (21.9 - 67.5i)T^{2}
73 1+(−1.09−3.36i)T+(−59.0+42.9i)T2 1 + (-1.09 - 3.36i)T + (-59.0 + 42.9i)T^{2}
79 1+(−0.541−0.393i)T+(24.4+75.1i)T2 1 + (-0.541 - 0.393i)T + (24.4 + 75.1i)T^{2}
83 1+(−10.8+7.89i)T+(25.6−78.9i)T2 1 + (-10.8 + 7.89i)T + (25.6 - 78.9i)T^{2}
89 1+16.6T+89T2 1 + 16.6T + 89T^{2}
97 1+(8.18+5.94i)T+(29.9+92.2i)T2 1 + (8.18 + 5.94i)T + (29.9 + 92.2i)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

−9.916921246119873881884918585926, −9.604903369872183652021395598878, −8.896868223333820564106963487790, −8.118065229101272692107462559413, −6.46926367119918212799016704575, −5.87416278860266605612519443467, −5.03432377327897480335653474978, −3.80841958972737565493144475825, −1.90624763737157694056633478086, −1.29969902137028553334864517190, 1.30537755849473125616633297092, 3.14629887081456028565227619477, 3.74358575587652294465101539353, 5.40870005532255801214229598385, 6.39596994044604731786996918108, 6.95691092737243736120831128571, 8.160915595911150230607768805887, 8.737635349656669039334618108167, 9.679445560987326568134426414796, 10.52726913100479488246153696953

Graph of the ZZ-function along the critical line