Properties

Label 2-4026-1.1-c1-0-58
Degree $2$
Conductor $4026$
Sign $1$
Analytic cond. $32.1477$
Root an. cond. $5.66990$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 1.48·5-s − 6-s + 4.53·7-s − 8-s + 9-s − 1.48·10-s + 11-s + 12-s + 3.20·13-s − 4.53·14-s + 1.48·15-s + 16-s + 1.19·17-s − 18-s + 7.44·19-s + 1.48·20-s + 4.53·21-s − 22-s − 6.57·23-s − 24-s − 2.78·25-s − 3.20·26-s + 27-s + 4.53·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.666·5-s − 0.408·6-s + 1.71·7-s − 0.353·8-s + 0.333·9-s − 0.470·10-s + 0.301·11-s + 0.288·12-s + 0.890·13-s − 1.21·14-s + 0.384·15-s + 0.250·16-s + 0.290·17-s − 0.235·18-s + 1.70·19-s + 0.333·20-s + 0.989·21-s − 0.213·22-s − 1.36·23-s − 0.204·24-s − 0.556·25-s − 0.629·26-s + 0.192·27-s + 0.856·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4026\)    =    \(2 \cdot 3 \cdot 11 \cdot 61\)
Sign: $1$
Analytic conductor: \(32.1477\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4026,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.821389925\)
\(L(\frac12)\) \(\approx\) \(2.821389925\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
11 \( 1 - T \)
61 \( 1 - T \)
good5 \( 1 - 1.48T + 5T^{2} \)
7 \( 1 - 4.53T + 7T^{2} \)
13 \( 1 - 3.20T + 13T^{2} \)
17 \( 1 - 1.19T + 17T^{2} \)
19 \( 1 - 7.44T + 19T^{2} \)
23 \( 1 + 6.57T + 23T^{2} \)
29 \( 1 - 2.91T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 + 3.44T + 37T^{2} \)
41 \( 1 + 4.39T + 41T^{2} \)
43 \( 1 + 3.88T + 43T^{2} \)
47 \( 1 + 4.10T + 47T^{2} \)
53 \( 1 - 2.64T + 53T^{2} \)
59 \( 1 - 0.254T + 59T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 - 5.70T + 71T^{2} \)
73 \( 1 + 2.33T + 73T^{2} \)
79 \( 1 + 3.35T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 - 5.69T + 89T^{2} \)
97 \( 1 + 3.32T + 97T^{2} \)
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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

−8.322733022048092822055562332826, −8.012328341162186646110415839005, −7.25041914866348853964323679286, −6.28222593758844875447696043380, −5.54345258701095046898118604675, −4.71899546961488399043319562620, −3.74335561068927480959165720323, −2.70657597138877180397386652621, −1.66585520809192549930232856215, −1.23765001500018875281274284957, 1.23765001500018875281274284957, 1.66585520809192549930232856215, 2.70657597138877180397386652621, 3.74335561068927480959165720323, 4.71899546961488399043319562620, 5.54345258701095046898118604675, 6.28222593758844875447696043380, 7.25041914866348853964323679286, 8.012328341162186646110415839005, 8.322733022048092822055562332826

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