Properties

Label 2-72450-1.1-c1-0-41
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 + 4·13-s + 14-s + 16-s − 4·17-s − 2·19-s − 4·22-s + 23-s − 4·26-s − 28-s + 6·29-s − 4·31-s − 32-s + 4·34-s + 2·37-s + 2·38-s + 10·41-s + 12·43-s + 4·44-s − 46-s + 49-s + 4·52-s + 4·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.20·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.458·19-s − 0.852·22-s + 0.208·23-s − 0.784·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.685·34-s + 0.328·37-s + 0.324·38-s + 1.56·41-s + 1.82·43-s + 0.603·44-s − 0.147·46-s + 1/7·49-s + 0.554·52-s + 0.549·53-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

\(L(1)\) \(\approx\) \(2.287515331\)
\(L(\frac12)\) \(\approx\) \(2.287515331\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.25071516600223, −13.59467458961403, −13.00336350269565, −12.63381107180749, −12.02144288440771, −11.46182207446263, −10.98508234272212, −10.71111058374552, −10.01895285713103, −9.332544662692147, −9.083282244679309, −8.645756792429821, −8.076087232554013, −7.423608697321919, −6.790508532734258, −6.406862404478243, −6.028558237398396, −5.306044779360149, −4.361559227882858, −3.976307017347730, −3.406291675134212, −2.495757868190875, −2.056187861801554, −1.043906006114946, −0.6855124637797767, 0.6855124637797767, 1.043906006114946, 2.056187861801554, 2.495757868190875, 3.406291675134212, 3.976307017347730, 4.361559227882858, 5.306044779360149, 6.028558237398396, 6.406862404478243, 6.790508532734258, 7.423608697321919, 8.076087232554013, 8.645756792429821, 9.083282244679309, 9.332544662692147, 10.01895285713103, 10.71111058374552, 10.98508234272212, 11.46182207446263, 12.02144288440771, 12.63381107180749, 13.00336350269565, 13.59467458961403, 14.25071516600223

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