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Properties

Label 2-72450-1.1-c1-0-11
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 − 4·11-s − 6·13-s + 14-s + 16-s + 2·17-s + 4·19-s − 4·22-s − 23-s − 6·26-s + 28-s + 2·29-s − 8·31-s + 32-s + 2·34-s + 2·37-s + 4·38-s + 6·41-s + 4·43-s − 4·44-s − 46-s + 49-s − 6·52-s − 2·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.852·22-s − 0.208·23-s − 1.17·26-s + 0.188·28-s + 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.328·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s − 0.603·44-s − 0.147·46-s + 1/7·49-s − 0.832·52-s − 0.274·53-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

\(L(1)\) \(\approx\) \(2.655350056\)
\(L(\frac12)\) \(\approx\) \(2.655350056\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−14.25690995538161, −13.68599443676761, −12.93927653865273, −12.69511915965782, −12.22623398160466, −11.66289797094748, −11.22932298155635, −10.51544189299122, −10.27626662667565, −9.526495857260355, −9.225798452472379, −8.280376989394994, −7.705602055194893, −7.481603041047301, −7.013295142609474, −6.134891267351468, −5.481284295986987, −5.276104870100602, −4.639566696184155, −4.138984743152791, −3.250528992753402, −2.763177997811472, −2.249465931008258, −1.484946006130711, −0.4579961174317779, 0.4579961174317779, 1.484946006130711, 2.249465931008258, 2.763177997811472, 3.250528992753402, 4.138984743152791, 4.639566696184155, 5.276104870100602, 5.481284295986987, 6.134891267351468, 7.013295142609474, 7.481603041047301, 7.705602055194893, 8.280376989394994, 9.225798452472379, 9.526495857260355, 10.27626662667565, 10.51544189299122, 11.22932298155635, 11.66289797094748, 12.22623398160466, 12.69511915965782, 12.93927653865273, 13.68599443676761, 14.25690995538161

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