Properties

Label 1-424-424.211-r1-0-0
Degree $1$
Conductor $424$
Sign $1$
Analytic cond. $45.5651$
Root an. cond. $45.5651$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 7-s + 9-s + 11-s − 13-s − 15-s + 17-s − 19-s + 21-s + 23-s + 25-s − 27-s − 29-s + 31-s − 33-s − 35-s − 37-s + 39-s − 41-s + 43-s + 45-s − 47-s + 49-s − 51-s + 55-s + 57-s + ⋯
L(s)  = 1  − 3-s + 5-s − 7-s + 9-s + 11-s − 13-s − 15-s + 17-s − 19-s + 21-s + 23-s + 25-s − 27-s − 29-s + 31-s − 33-s − 35-s − 37-s + 39-s − 41-s + 43-s + 45-s − 47-s + 49-s − 51-s + 55-s + 57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 424 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 424 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(424\)    =    \(2^{3} \cdot 53\)
Sign: $1$
Analytic conductor: \(45.5651\)
Root analytic conductor: \(45.5651\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{424} (211, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 424,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.481281234\)
\(L(\frac12)\) \(\approx\) \(1.481281234\)
\(L(1)\) \(\approx\) \(0.9154153589\)
\(L(1)\) \(\approx\) \(0.9154153589\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
53 \( 1 \)
good3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.96888460531913007548432980925, −22.78480453794951728256857742849, −22.414091678821967422900160513735, −21.568173638551068346564944417925, −20.79926550592624244791464132401, −19.32403647357346698108561730700, −18.893779763093633418871704168348, −17.59486115403866407774290004796, −16.96837166668023976456315485140, −16.556609893304218336973759162921, −15.23668993712284977427449321831, −14.302529864094538691319969669772, −13.11468723400460372739919390419, −12.52462221232568897398710952888, −11.609029703297483630306712027829, −10.3481270090966121857877715245, −9.83704753448865161300940486941, −8.94638058710642936311029775512, −7.1797470860939879802714314626, −6.5095970188772512616110408869, −5.700528689387835237822497539653, −4.73127730230385339505272769796, −3.39879079009920655466502819786, −1.97803000720607482878202534423, −0.72015020256434764667459459696, 0.72015020256434764667459459696, 1.97803000720607482878202534423, 3.39879079009920655466502819786, 4.73127730230385339505272769796, 5.700528689387835237822497539653, 6.5095970188772512616110408869, 7.1797470860939879802714314626, 8.94638058710642936311029775512, 9.83704753448865161300940486941, 10.3481270090966121857877715245, 11.609029703297483630306712027829, 12.52462221232568897398710952888, 13.11468723400460372739919390419, 14.302529864094538691319969669772, 15.23668993712284977427449321831, 16.556609893304218336973759162921, 16.96837166668023976456315485140, 17.59486115403866407774290004796, 18.893779763093633418871704168348, 19.32403647357346698108561730700, 20.79926550592624244791464132401, 21.568173638551068346564944417925, 22.414091678821967422900160513735, 22.78480453794951728256857742849, 23.96888460531913007548432980925

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