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Properties

Label 2-72450-1.1-c1-0-28
Degree $2$
Conductor $72450$
Sign $1$
Analytic cond. $578.516$
Root an. cond. $24.0523$
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 − 7-s + 8-s − 6·11-s + 4·13-s − 14-s + 16-s − 2·17-s + 4·19-s − 6·22-s + 23-s + 4·26-s − 28-s + 10·29-s − 8·31-s + 32-s − 2·34-s + 8·37-s + 4·38-s + 2·41-s − 6·43-s − 6·44-s + 46-s + 12·47-s + 49-s + 4·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.80·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 1.27·22-s + 0.208·23-s + 0.784·26-s − 0.188·28-s + 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s + 1.31·37-s + 0.648·38-s + 0.312·41-s − 0.914·43-s − 0.904·44-s + 0.147·46-s + 1.75·47-s + 1/7·49-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(578.516\)
Root analytic conductor: \(24.0523\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.501983458\)
\(L(\frac12)\) \(\approx\) \(3.501983458\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.91351725904030, −13.53199440508406, −13.08591341276944, −12.90900161265934, −12.06694577779244, −11.77165077925892, −10.99055777926940, −10.69765219104020, −10.23505492727273, −9.697422164416887, −8.819780210445368, −8.603316258927548, −7.729477058346721, −7.476172598845248, −6.807711998401770, −6.163540800386342, −5.675647862539791, −5.242036719827523, −4.613868517577453, −3.988187067203678, −3.354058431819845, −2.708319265735615, −2.388073899292614, −1.325835870480955, −0.5636174419208153, 0.5636174419208153, 1.325835870480955, 2.388073899292614, 2.708319265735615, 3.354058431819845, 3.988187067203678, 4.613868517577453, 5.242036719827523, 5.675647862539791, 6.163540800386342, 6.807711998401770, 7.476172598845248, 7.729477058346721, 8.603316258927548, 8.819780210445368, 9.697422164416887, 10.23505492727273, 10.69765219104020, 10.99055777926940, 11.77165077925892, 12.06694577779244, 12.90900161265934, 13.08591341276944, 13.53199440508406, 13.91351725904030

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