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Properties

Label 2-8046-1.1-c1-0-154
Degree $2$
Conductor $8046$
Sign $-1$
Analytic cond. $64.2476$
Root an. cond. $8.01546$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2·5-s + 7-s + 8-s − 2·10-s − 2·13-s + 14-s + 16-s − 4·17-s + 4·19-s − 2·20-s + 5·23-s − 25-s − 2·26-s + 28-s − 29-s − 5·31-s + 32-s − 4·34-s − 2·35-s − 6·37-s + 4·38-s − 2·40-s + 9·41-s − 4·43-s + 5·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.917·19-s − 0.447·20-s + 1.04·23-s − 1/5·25-s − 0.392·26-s + 0.188·28-s − 0.185·29-s − 0.898·31-s + 0.176·32-s − 0.685·34-s − 0.338·35-s − 0.986·37-s + 0.648·38-s − 0.316·40-s + 1.40·41-s − 0.609·43-s + 0.737·46-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8046,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.37953027768837106783962470187, −6.94902811346530516535084185164, −6.00349519945841203064216099872, −5.21505028067511115441520100545, −4.66377695270198579742645853657, −3.95932542186001261708726753823, −3.24081105297797690749435891521, −2.41580478207842296073782829612, −1.37854838140684239127386514280, 0, 1.37854838140684239127386514280, 2.41580478207842296073782829612, 3.24081105297797690749435891521, 3.95932542186001261708726753823, 4.66377695270198579742645853657, 5.21505028067511115441520100545, 6.00349519945841203064216099872, 6.94902811346530516535084185164, 7.37953027768837106783962470187

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