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Properties

Label 2-72450-1.1-c1-0-16
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 + 13-s + 14-s + 16-s − 3·17-s − 4·19-s − 6·22-s − 23-s − 26-s − 28-s + 6·29-s + 8·31-s − 32-s + 3·34-s − 11·37-s + 4·38-s − 8·43-s + 6·44-s + 46-s + 3·47-s + 49-s + 52-s + 9·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.80·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.917·19-s − 1.27·22-s − 0.208·23-s − 0.196·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.514·34-s − 1.80·37-s + 0.648·38-s − 1.21·43-s + 0.904·44-s + 0.147·46-s + 0.437·47-s + 1/7·49-s + 0.138·52-s + 1.23·53-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

\(L(1)\) \(\approx\) \(1.583324891\)
\(L(\frac12)\) \(\approx\) \(1.583324891\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.10823069041978, −13.63783370856498, −13.16855803554048, −12.35519663133900, −12.02609672688109, −11.66141213329174, −11.07180460625177, −10.41654873873039, −10.10125351884021, −9.533778418460839, −8.891011167689959, −8.544714084880942, −8.310662014268025, −7.252798262360301, −6.819408516199462, −6.466716255551314, −6.069996453559348, −5.210587650927883, −4.422849453594096, −3.984564949453527, −3.348823174747481, −2.609985014542833, −1.882558745556683, −1.267362191938150, −0.4956034480445889, 0.4956034480445889, 1.267362191938150, 1.882558745556683, 2.609985014542833, 3.348823174747481, 3.984564949453527, 4.422849453594096, 5.210587650927883, 6.069996453559348, 6.466716255551314, 6.819408516199462, 7.252798262360301, 8.310662014268025, 8.544714084880942, 8.891011167689959, 9.533778418460839, 10.10125351884021, 10.41654873873039, 11.07180460625177, 11.66141213329174, 12.02609672688109, 12.35519663133900, 13.16855803554048, 13.63783370856498, 14.10823069041978

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