Properties

Label 2-32490-1.1-c1-0-24
Degree $2$
Conductor $32490$
Sign $1$
Analytic cond. $259.433$
Root an. cond. $16.1069$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s + 2·13-s + 2·14-s + 16-s + 6·17-s + 20-s − 6·23-s + 25-s + 2·26-s + 2·28-s + 4·29-s + 32-s + 6·34-s + 2·35-s + 10·37-s + 40-s + 8·41-s + 2·43-s − 6·46-s + 2·47-s − 3·49-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.377·28-s + 0.742·29-s + 0.176·32-s + 1.02·34-s + 0.338·35-s + 1.64·37-s + 0.158·40-s + 1.24·41-s + 0.304·43-s − 0.884·46-s + 0.291·47-s − 3/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32490\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(259.433\)
Root analytic conductor: \(16.1069\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{32490} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32490,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.859220833\)
\(L(\frac12)\) \(\approx\) \(5.859220833\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
19 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.73856062366215, −14.42006082379185, −14.17637558910259, −13.46690572155743, −13.01726844874496, −12.39530717490159, −11.93366220534119, −11.38831071551781, −10.91709376304980, −10.20526449501392, −9.856585667047440, −9.162043634502828, −8.335925856883815, −7.935484257366253, −7.457230220535878, −6.602582529513825, −6.028803304773677, −5.600410504955872, −5.036895912441907, −4.222109218506016, −3.864172856912432, −2.913437299835938, −2.369347658615404, −1.490919749121842, −0.8810690245203166, 0.8810690245203166, 1.490919749121842, 2.369347658615404, 2.913437299835938, 3.864172856912432, 4.222109218506016, 5.036895912441907, 5.600410504955872, 6.028803304773677, 6.602582529513825, 7.457230220535878, 7.935484257366253, 8.335925856883815, 9.162043634502828, 9.856585667047440, 10.20526449501392, 10.91709376304980, 11.38831071551781, 11.93366220534119, 12.39530717490159, 13.01726844874496, 13.46690572155743, 14.17637558910259, 14.42006082379185, 14.73856062366215

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