Properties

Label 2-4027-1.1-c1-0-9
Degree $2$
Conductor $4027$
Sign $1$
Analytic cond. $32.1557$
Root an. cond. $5.67060$
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  − 2.37·2-s − 3.14·3-s + 3.64·4-s + 0.257·5-s + 7.48·6-s + 1.36·7-s − 3.90·8-s + 6.91·9-s − 0.611·10-s − 3.32·11-s − 11.4·12-s − 0.510·13-s − 3.23·14-s − 0.810·15-s + 1.98·16-s − 5.42·17-s − 16.4·18-s − 5.24·19-s + 0.938·20-s − 4.28·21-s + 7.90·22-s + 1.00·23-s + 12.2·24-s − 4.93·25-s + 1.21·26-s − 12.3·27-s + 4.96·28-s + ⋯
L(s)  = 1  − 1.67·2-s − 1.81·3-s + 1.82·4-s + 0.115·5-s + 3.05·6-s + 0.514·7-s − 1.37·8-s + 2.30·9-s − 0.193·10-s − 1.00·11-s − 3.31·12-s − 0.141·13-s − 0.864·14-s − 0.209·15-s + 0.496·16-s − 1.31·17-s − 3.87·18-s − 1.20·19-s + 0.209·20-s − 0.936·21-s + 1.68·22-s + 0.208·23-s + 2.50·24-s − 0.986·25-s + 0.237·26-s − 2.37·27-s + 0.937·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4027\)
Sign: $1$
Analytic conductor: \(32.1557\)
Root analytic conductor: \(5.67060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4027,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06387245058\)
\(L(\frac12)\) \(\approx\) \(0.06387245058\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4027 \( 1+O(T) \)
good2 \( 1 + 2.37T + 2T^{2} \)
3 \( 1 + 3.14T + 3T^{2} \)
5 \( 1 - 0.257T + 5T^{2} \)
7 \( 1 - 1.36T + 7T^{2} \)
11 \( 1 + 3.32T + 11T^{2} \)
13 \( 1 + 0.510T + 13T^{2} \)
17 \( 1 + 5.42T + 17T^{2} \)
19 \( 1 + 5.24T + 19T^{2} \)
23 \( 1 - 1.00T + 23T^{2} \)
29 \( 1 + 1.81T + 29T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + 1.60T + 37T^{2} \)
41 \( 1 + 7.60T + 41T^{2} \)
43 \( 1 + 5.13T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 - 9.39T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 + 8.61T + 61T^{2} \)
67 \( 1 + 5.63T + 67T^{2} \)
71 \( 1 + 3.79T + 71T^{2} \)
73 \( 1 - 2.36T + 73T^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + 2.46T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 - 4.70T + 97T^{2} \)
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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

−8.508502312794903181847381223889, −7.65012217559226135791784672689, −7.05771872616748512771837638242, −6.46147946437348788361799381826, −5.63471115803788385784884677269, −4.96646465528633399501909433322, −4.06790340196555556513909505481, −2.23039780444699698599129598353, −1.61443933412807182200637316357, −0.20275899913021850455650201835, 0.20275899913021850455650201835, 1.61443933412807182200637316357, 2.23039780444699698599129598353, 4.06790340196555556513909505481, 4.96646465528633399501909433322, 5.63471115803788385784884677269, 6.46147946437348788361799381826, 7.05771872616748512771837638242, 7.65012217559226135791784672689, 8.508502312794903181847381223889

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