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Properties

Label 2-651-651.17-c0-0-0
Degree $2$
Conductor $651$
Sign $0.895 - 0.444i$
Analytic cond. $0.324891$
Root an. cond. $0.569992$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)3-s + (0.913 − 0.406i)4-s + (−0.669 − 0.743i)7-s + (−0.809 + 0.587i)9-s + (0.669 + 0.743i)12-s + (1.78 + 0.379i)13-s + (0.669 − 0.743i)16-s + (−0.873 + 0.786i)19-s + (0.499 − 0.866i)21-s − 25-s + (−0.809 − 0.587i)27-s + (−0.913 − 0.406i)28-s + (−0.669 + 0.743i)31-s + (−0.499 + 0.866i)36-s + (−1.28 − 0.743i)37-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)3-s + (0.913 − 0.406i)4-s + (−0.669 − 0.743i)7-s + (−0.809 + 0.587i)9-s + (0.669 + 0.743i)12-s + (1.78 + 0.379i)13-s + (0.669 − 0.743i)16-s + (−0.873 + 0.786i)19-s + (0.499 − 0.866i)21-s − 25-s + (−0.809 − 0.587i)27-s + (−0.913 − 0.406i)28-s + (−0.669 + 0.743i)31-s + (−0.499 + 0.866i)36-s + (−1.28 − 0.743i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 651 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 651 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(651\)    =    \(3 \cdot 7 \cdot 31\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(0.324891\)
Root analytic conductor: \(0.569992\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{651} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 651,\ (\ :0),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.166294535\)
\(L(\frac12)\) \(\approx\) \(1.166294535\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
good2 \( 1 + (-0.913 + 0.406i)T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + (-0.978 - 0.207i)T^{2} \)
13 \( 1 + (-1.78 - 0.379i)T + (0.913 + 0.406i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.873 - 0.786i)T + (0.104 - 0.994i)T^{2} \)
23 \( 1 + (0.669 + 0.743i)T^{2} \)
29 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (1.28 + 0.743i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.104 + 0.994i)T^{2} \)
43 \( 1 + (0.395 + 1.86i)T + (-0.913 + 0.406i)T^{2} \)
47 \( 1 + (-0.104 - 0.994i)T^{2} \)
53 \( 1 + (0.669 + 0.743i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.104 + 0.181i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.978 - 0.207i)T^{2} \)
73 \( 1 + (-0.204 + 1.94i)T + (-0.978 - 0.207i)T^{2} \)
79 \( 1 + (0.809 - 0.0850i)T + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (0.104 - 0.994i)T^{2} \)
89 \( 1 + (0.978 + 0.207i)T^{2} \)
97 \( 1 + (1.16 - 1.60i)T + (-0.309 - 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.54907361367706112705351074338, −10.31196615235233710695441702944, −9.170809425572223911040841038868, −8.367021055849427407963436640893, −7.21372275928861350528627662158, −6.26646785314415110257835921438, −5.52882734620183348708457211149, −3.97336186919728612823989764482, −3.43964702152219192528045791733, −1.87042952854743141367267594926, 1.74363270228082841656787205857, 2.84712714102701571400045086997, 3.70168269131654433246116271288, 5.75023306093386062072835403998, 6.32570601936882436542067710095, 7.03646161919591043870517873125, 8.221716375336499466052333075636, 8.585799480017565395844140426136, 9.749365299863764681821334416193, 11.08892267730122125113863702422

Graph of the $Z$-function along the critical line