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Properties

Label 2-693-11.3-c1-0-21
Degree 22
Conductor 693693
Sign 0.642+0.766i0.642 + 0.766i
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  + (−0.5 + 0.363i)2-s + (−0.5 + 1.53i)4-s + (−0.309 − 0.224i)5-s + (0.309 − 0.951i)7-s + (−0.690 − 2.12i)8-s + 0.236·10-s + (−0.309 − 3.30i)11-s + (0.809 − 0.587i)13-s + (0.190 + 0.587i)14-s + (−1.49 − 1.08i)16-s + (−3.42 − 2.48i)17-s + (0.5 − 0.363i)20-s + (1.35 + 1.53i)22-s + 3.23·23-s + (−1.5 − 4.61i)25-s + (−0.190 + 0.587i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.256i)2-s + (−0.250 + 0.769i)4-s + (−0.138 − 0.100i)5-s + (0.116 − 0.359i)7-s + (−0.244 − 0.751i)8-s + 0.0746·10-s + (−0.0931 − 0.995i)11-s + (0.224 − 0.163i)13-s + (0.0510 + 0.157i)14-s + (−0.374 − 0.272i)16-s + (−0.831 − 0.603i)17-s + (0.111 − 0.0812i)20-s + (0.288 + 0.328i)22-s + 0.674·23-s + (−0.300 − 0.923i)25-s + (−0.0374 + 0.115i)26-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((0.642+0.766i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=((0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 0.642+0.766i0.642 + 0.766i
Analytic conductor: 5.533635.53363
Root analytic conductor: 2.352362.35236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ693(190,⋅)\chi_{693} (190, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :1/2), 0.642+0.766i)(2,\ 693,\ (\ :1/2),\ 0.642 + 0.766i)

Particular Values

L(1)L(1) ≈\approx 0.800948−0.373624i0.800948 - 0.373624i
L(12)L(\frac12) ≈\approx 0.800948−0.373624i0.800948 - 0.373624i
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+(0.309+3.30i)T 1 + (0.309 + 3.30i)T
good2 1+(0.5−0.363i)T+(0.618−1.90i)T2 1 + (0.5 - 0.363i)T + (0.618 - 1.90i)T^{2}
5 1+(0.309+0.224i)T+(1.54+4.75i)T2 1 + (0.309 + 0.224i)T + (1.54 + 4.75i)T^{2}
13 1+(−0.809+0.587i)T+(4.01−12.3i)T2 1 + (-0.809 + 0.587i)T + (4.01 - 12.3i)T^{2}
17 1+(3.42+2.48i)T+(5.25+16.1i)T2 1 + (3.42 + 2.48i)T + (5.25 + 16.1i)T^{2}
19 1+(−15.3+11.1i)T2 1 + (-15.3 + 11.1i)T^{2}
23 1−3.23T+23T2 1 - 3.23T + 23T^{2}
29 1+(2.07−6.37i)T+(−23.4−17.0i)T2 1 + (2.07 - 6.37i)T + (-23.4 - 17.0i)T^{2}
31 1+(−8.28+6.01i)T+(9.57−29.4i)T2 1 + (-8.28 + 6.01i)T + (9.57 - 29.4i)T^{2}
37 1+(−2.14+6.60i)T+(−29.9−21.7i)T2 1 + (-2.14 + 6.60i)T + (-29.9 - 21.7i)T^{2}
41 1+(1.57+4.84i)T+(−33.1+24.0i)T2 1 + (1.57 + 4.84i)T + (-33.1 + 24.0i)T^{2}
43 1+T+43T2 1 + T + 43T^{2}
47 1+(−2.26−6.96i)T+(−38.0+27.6i)T2 1 + (-2.26 - 6.96i)T + (-38.0 + 27.6i)T^{2}
53 1+(−6.16+4.47i)T+(16.3−50.4i)T2 1 + (-6.16 + 4.47i)T + (16.3 - 50.4i)T^{2}
59 1+(−1.28+3.94i)T+(−47.7−34.6i)T2 1 + (-1.28 + 3.94i)T + (-47.7 - 34.6i)T^{2}
61 1+(−4.66−3.38i)T+(18.8+58.0i)T2 1 + (-4.66 - 3.38i)T + (18.8 + 58.0i)T^{2}
67 1+9.23T+67T2 1 + 9.23T + 67T^{2}
71 1+(6.04+4.39i)T+(21.9+67.5i)T2 1 + (6.04 + 4.39i)T + (21.9 + 67.5i)T^{2}
73 1+(−3.57+10.9i)T+(−59.0−42.9i)T2 1 + (-3.57 + 10.9i)T + (-59.0 - 42.9i)T^{2}
79 1+(8.78−6.37i)T+(24.4−75.1i)T2 1 + (8.78 - 6.37i)T + (24.4 - 75.1i)T^{2}
83 1+(4.85+3.52i)T+(25.6+78.9i)T2 1 + (4.85 + 3.52i)T + (25.6 + 78.9i)T^{2}
89 1−6.38T+89T2 1 - 6.38T + 89T^{2}
97 1+(−13.7+9.99i)T+(29.9−92.2i)T2 1 + (-13.7 + 9.99i)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

−10.31790855487278424014791507338, −9.188401936647555327318239001993, −8.625243314756290111304122599665, −7.81243431599229907677910197323, −7.01032964050144868795948051171, −6.03304673580676565245667378003, −4.71411613428950864728825921064, −3.78193748237233534474475280577, −2.70461031854597476297427667824, −0.55808287327551655038774660347, 1.45107482186004068518253386746, 2.60042228517295993933452035322, 4.24801934825283045988662266753, 5.08852138783178100142832237390, 6.13515488110428919791766209177, 7.02004833729126138474054788288, 8.239682595456951553985256133482, 8.941358872247539193325722400329, 9.827449566951611664492266378664, 10.41353926471982110834346376509

Graph of the ZZ-function along the critical line