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Properties

Label 2-693-11.3-c1-0-29
Degree 22
Conductor 693693
Sign −0.993+0.110i-0.993 + 0.110i
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  + (1.99 − 1.44i)2-s + (1.26 − 3.88i)4-s + (−2.80 − 2.03i)5-s + (−0.309 + 0.951i)7-s + (−1.58 − 4.89i)8-s − 8.55·10-s + (−2.91 − 1.57i)11-s + (0.528 − 0.384i)13-s + (0.762 + 2.34i)14-s + (−3.65 − 2.65i)16-s + (−0.919 − 0.668i)17-s + (−1.87 − 5.77i)19-s + (−11.4 + 8.32i)20-s + (−8.10 + 1.09i)22-s + 6.66·23-s + ⋯
L(s)  = 1  + (1.41 − 1.02i)2-s + (0.631 − 1.94i)4-s + (−1.25 − 0.911i)5-s + (−0.116 + 0.359i)7-s + (−0.561 − 1.72i)8-s − 2.70·10-s + (−0.880 − 0.474i)11-s + (0.146 − 0.106i)13-s + (0.203 + 0.626i)14-s + (−0.913 − 0.663i)16-s + (−0.223 − 0.162i)17-s + (−0.430 − 1.32i)19-s + (−2.56 + 1.86i)20-s + (−1.72 + 0.233i)22-s + 1.39·23-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((−0.993+0.110i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=((−0.993+0.110i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: −0.993+0.110i-0.993 + 0.110i
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), −0.993+0.110i)(2,\ 693,\ (\ :1/2),\ -0.993 + 0.110i)

Particular Values

L(1)L(1) ≈\approx 0.119979−2.17002i0.119979 - 2.17002i
L(12)L(\frac12) ≈\approx 0.119979−2.17002i0.119979 - 2.17002i
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+(2.91+1.57i)T 1 + (2.91 + 1.57i)T
good2 1+(−1.99+1.44i)T+(0.618−1.90i)T2 1 + (-1.99 + 1.44i)T + (0.618 - 1.90i)T^{2}
5 1+(2.80+2.03i)T+(1.54+4.75i)T2 1 + (2.80 + 2.03i)T + (1.54 + 4.75i)T^{2}
13 1+(−0.528+0.384i)T+(4.01−12.3i)T2 1 + (-0.528 + 0.384i)T + (4.01 - 12.3i)T^{2}
17 1+(0.919+0.668i)T+(5.25+16.1i)T2 1 + (0.919 + 0.668i)T + (5.25 + 16.1i)T^{2}
19 1+(1.87+5.77i)T+(−15.3+11.1i)T2 1 + (1.87 + 5.77i)T + (-15.3 + 11.1i)T^{2}
23 1−6.66T+23T2 1 - 6.66T + 23T^{2}
29 1+(−1.41+4.34i)T+(−23.4−17.0i)T2 1 + (-1.41 + 4.34i)T + (-23.4 - 17.0i)T^{2}
31 1+(2.26−1.64i)T+(9.57−29.4i)T2 1 + (2.26 - 1.64i)T + (9.57 - 29.4i)T^{2}
37 1+(0.135−0.418i)T+(−29.9−21.7i)T2 1 + (0.135 - 0.418i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.82−5.61i)T+(−33.1+24.0i)T2 1 + (-1.82 - 5.61i)T + (-33.1 + 24.0i)T^{2}
43 1−8.70T+43T2 1 - 8.70T + 43T^{2}
47 1+(−0.186−0.575i)T+(−38.0+27.6i)T2 1 + (-0.186 - 0.575i)T + (-38.0 + 27.6i)T^{2}
53 1+(−7.94+5.77i)T+(16.3−50.4i)T2 1 + (-7.94 + 5.77i)T + (16.3 - 50.4i)T^{2}
59 1+(0.523−1.61i)T+(−47.7−34.6i)T2 1 + (0.523 - 1.61i)T + (-47.7 - 34.6i)T^{2}
61 1+(5.54+4.03i)T+(18.8+58.0i)T2 1 + (5.54 + 4.03i)T + (18.8 + 58.0i)T^{2}
67 1+6.17T+67T2 1 + 6.17T + 67T^{2}
71 1+(4.38+3.18i)T+(21.9+67.5i)T2 1 + (4.38 + 3.18i)T + (21.9 + 67.5i)T^{2}
73 1+(−2.07+6.37i)T+(−59.0−42.9i)T2 1 + (-2.07 + 6.37i)T + (-59.0 - 42.9i)T^{2}
79 1+(2.14−1.55i)T+(24.4−75.1i)T2 1 + (2.14 - 1.55i)T + (24.4 - 75.1i)T^{2}
83 1+(−5.41−3.93i)T+(25.6+78.9i)T2 1 + (-5.41 - 3.93i)T + (25.6 + 78.9i)T^{2}
89 1−0.698T+89T2 1 - 0.698T + 89T^{2}
97 1+(−12.0+8.73i)T+(29.9−92.2i)T2 1 + (-12.0 + 8.73i)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

−10.57946830744356126212923265588, −9.227682530114262704464120121077, −8.440841039738486025761312208183, −7.35145262198676237215281654312, −6.04537737810462681825168430192, −4.97841851917790828685264431259, −4.55213339203988183233421395176, −3.42056539807196959003015278642, −2.53480934452181999891785586063, −0.72533052911107032105685989721, 2.77974087146392383975292270184, 3.71409200889205907212782224593, 4.38249352175883858580943537710, 5.47938127583271953514765118478, 6.48971547976662497083894798769, 7.40776487684974797565878636455, 7.58453044647223899253709286626, 8.732443189655097592092963263833, 10.40262453769198955292205392286, 10.97322453602633590430554953858

Graph of the ZZ-function along the critical line