Properties

Label 2-8046-1.1-c1-0-141
Degree $2$
Conductor $8046$
Sign $1$
Analytic cond. $64.2476$
Root an. cond. $8.01546$
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 + 4-s + 3.46·5-s + 4.46·7-s + 8-s + 3.46·10-s + 4.73·11-s − 1.46·13-s + 4.46·14-s + 16-s − 3.26·17-s + 1.46·19-s + 3.46·20-s + 4.73·22-s − 7.73·23-s + 6.99·25-s − 1.46·26-s + 4.46·28-s − 1.73·29-s + 31-s + 32-s − 3.26·34-s + 15.4·35-s − 0.732·37-s + 1.46·38-s + 3.46·40-s + 9.19·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.54·5-s + 1.68·7-s + 0.353·8-s + 1.09·10-s + 1.42·11-s − 0.406·13-s + 1.19·14-s + 0.250·16-s − 0.792·17-s + 0.335·19-s + 0.774·20-s + 1.00·22-s − 1.61·23-s + 1.39·25-s − 0.287·26-s + 0.843·28-s − 0.321·29-s + 0.179·31-s + 0.176·32-s − 0.560·34-s + 2.61·35-s − 0.120·37-s + 0.237·38-s + 0.547·40-s + 1.43·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8046\)    =    \(2 \cdot 3^{3} \cdot 149\)
Sign: $1$
Analytic conductor: \(64.2476\)
Root analytic conductor: \(8.01546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8046,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.150045430\)
\(L(\frac12)\) \(\approx\) \(6.150045430\)
\(L(\frac{3}{2})\) 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 \( 1 \)
149 \( 1 - T \)
good5 \( 1 - 3.46T + 5T^{2} \)
7 \( 1 - 4.46T + 7T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 + 3.26T + 17T^{2} \)
19 \( 1 - 1.46T + 19T^{2} \)
23 \( 1 + 7.73T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 0.732T + 37T^{2} \)
41 \( 1 - 9.19T + 41T^{2} \)
43 \( 1 + 4.92T + 43T^{2} \)
47 \( 1 + 0.535T + 47T^{2} \)
53 \( 1 + 9.73T + 53T^{2} \)
59 \( 1 - 6.73T + 59T^{2} \)
61 \( 1 + 0.732T + 61T^{2} \)
67 \( 1 - 4.19T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + 5.46T + 73T^{2} \)
79 \( 1 + 9.26T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 7.73T + 89T^{2} \)
97 \( 1 - 7.26T + 97T^{2} \)
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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

−7.72720891319110416624854034773, −7.00072880580872996460561852682, −6.13496630048480084751671968076, −5.87259821424477526384663648364, −4.91920035544676230131853337357, −4.51806151196447967538312361411, −3.67279716365511574786775073178, −2.38077471860875336811629424598, −1.89507983287568893425300357974, −1.26423001087736680828892328610, 1.26423001087736680828892328610, 1.89507983287568893425300357974, 2.38077471860875336811629424598, 3.67279716365511574786775073178, 4.51806151196447967538312361411, 4.91920035544676230131853337357, 5.87259821424477526384663648364, 6.13496630048480084751671968076, 7.00072880580872996460561852682, 7.72720891319110416624854034773

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