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Properties

Label 2-693-11.3-c1-0-5
Degree 22
Conductor 693693
Sign −0.993−0.114i-0.993 - 0.114i
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  + (−2.14 + 1.55i)2-s + (1.55 − 4.78i)4-s + (1.38 + 1.00i)5-s + (−0.309 + 0.951i)7-s + (2.48 + 7.64i)8-s − 4.54·10-s + (−2.49 − 2.18i)11-s + (−1.22 + 0.888i)13-s + (−0.819 − 2.52i)14-s + (−9.11 − 6.62i)16-s + (−1.93 − 1.40i)17-s + (1.95 + 6.02i)19-s + (6.97 − 5.06i)20-s + (8.75 + 0.795i)22-s + 5.05·23-s + ⋯
L(s)  = 1  + (−1.51 + 1.10i)2-s + (0.777 − 2.39i)4-s + (0.619 + 0.450i)5-s + (−0.116 + 0.359i)7-s + (0.878 + 2.70i)8-s − 1.43·10-s + (−0.752 − 0.658i)11-s + (−0.339 + 0.246i)13-s + (−0.219 − 0.674i)14-s + (−2.27 − 1.65i)16-s + (−0.469 − 0.341i)17-s + (0.448 + 1.38i)19-s + (1.55 − 1.13i)20-s + (1.86 + 0.169i)22-s + 1.05·23-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((−0.993−0.114i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=((−0.993−0.114i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: −0.993−0.114i-0.993 - 0.114i
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), −0.993−0.114i)(2,\ 693,\ (\ :1/2),\ -0.993 - 0.114i)

Particular Values

L(1)L(1) ≈\approx 0.0277048+0.484239i0.0277048 + 0.484239i
L(12)L(\frac12) ≈\approx 0.0277048+0.484239i0.0277048 + 0.484239i
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+(2.49+2.18i)T 1 + (2.49 + 2.18i)T
good2 1+(2.14−1.55i)T+(0.618−1.90i)T2 1 + (2.14 - 1.55i)T + (0.618 - 1.90i)T^{2}
5 1+(−1.38−1.00i)T+(1.54+4.75i)T2 1 + (-1.38 - 1.00i)T + (1.54 + 4.75i)T^{2}
13 1+(1.22−0.888i)T+(4.01−12.3i)T2 1 + (1.22 - 0.888i)T + (4.01 - 12.3i)T^{2}
17 1+(1.93+1.40i)T+(5.25+16.1i)T2 1 + (1.93 + 1.40i)T + (5.25 + 16.1i)T^{2}
19 1+(−1.95−6.02i)T+(−15.3+11.1i)T2 1 + (-1.95 - 6.02i)T + (-15.3 + 11.1i)T^{2}
23 1−5.05T+23T2 1 - 5.05T + 23T^{2}
29 1+(2.57−7.93i)T+(−23.4−17.0i)T2 1 + (2.57 - 7.93i)T + (-23.4 - 17.0i)T^{2}
31 1+(4.88−3.54i)T+(9.57−29.4i)T2 1 + (4.88 - 3.54i)T + (9.57 - 29.4i)T^{2}
37 1+(2.30−7.10i)T+(−29.9−21.7i)T2 1 + (2.30 - 7.10i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.55−4.78i)T+(−33.1+24.0i)T2 1 + (-1.55 - 4.78i)T + (-33.1 + 24.0i)T^{2}
43 1−2.02T+43T2 1 - 2.02T + 43T^{2}
47 1+(−1.70−5.25i)T+(−38.0+27.6i)T2 1 + (-1.70 - 5.25i)T + (-38.0 + 27.6i)T^{2}
53 1+(−5.04+3.66i)T+(16.3−50.4i)T2 1 + (-5.04 + 3.66i)T + (16.3 - 50.4i)T^{2}
59 1+(3.45−10.6i)T+(−47.7−34.6i)T2 1 + (3.45 - 10.6i)T + (-47.7 - 34.6i)T^{2}
61 1+(11.1+8.11i)T+(18.8+58.0i)T2 1 + (11.1 + 8.11i)T + (18.8 + 58.0i)T^{2}
67 1−13.6T+67T2 1 - 13.6T + 67T^{2}
71 1+(11.5+8.42i)T+(21.9+67.5i)T2 1 + (11.5 + 8.42i)T + (21.9 + 67.5i)T^{2}
73 1+(0.441−1.36i)T+(−59.0−42.9i)T2 1 + (0.441 - 1.36i)T + (-59.0 - 42.9i)T^{2}
79 1+(5.43−3.94i)T+(24.4−75.1i)T2 1 + (5.43 - 3.94i)T + (24.4 - 75.1i)T^{2}
83 1+(−4.74−3.45i)T+(25.6+78.9i)T2 1 + (-4.74 - 3.45i)T + (25.6 + 78.9i)T^{2}
89 1+16.4T+89T2 1 + 16.4T + 89T^{2}
97 1+(9.74−7.08i)T+(29.9−92.2i)T2 1 + (9.74 - 7.08i)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

−10.56957304148033285749847602883, −9.790096196786967060379435814232, −9.082543208629966329201654054223, −8.331085875664552634015803808735, −7.43749536499936668432399915304, −6.67160343577078547522785343707, −5.82538681281014331393847203639, −5.12586466950483169183236472353, −2.90171120147881524531530674755, −1.52630953184281189018113277747, 0.41648676554171505217232410654, 1.90740630070246445279439066853, 2.77195769962929199992623669793, 4.16170152229583985852166384179, 5.47740694391306011887442459674, 7.12346215186847059051398192834, 7.54576265105133909252225010186, 8.728722856495720963202892042694, 9.349943149260382251145019524893, 9.917892359042027241805552815047

Graph of the ZZ-function along the critical line