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Properties

Label 2-72450-1.1-c1-0-9
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 + 14-s + 16-s + 6·17-s − 6·19-s + 4·22-s − 23-s − 28-s − 10·29-s + 4·31-s − 32-s − 6·34-s + 2·37-s + 6·38-s + 10·41-s + 4·43-s − 4·44-s + 46-s + 12·47-s + 49-s − 6·53-s + 56-s + 10·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.20·11-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.37·19-s + 0.852·22-s − 0.208·23-s − 0.188·28-s − 1.85·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.973·38-s + 1.56·41-s + 0.609·43-s − 0.603·44-s + 0.147·46-s + 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.133·56-s + 1.31·58-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

\(L(1)\) \(\approx\) \(0.9984696450\)
\(L(\frac12)\) \(\approx\) \(0.9984696450\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−14.37065047710329, −13.49260754554848, −12.94229889181452, −12.63745436258497, −12.19094019654634, −11.45678054506663, −10.92789742735676, −10.55477562750277, −10.07172748423433, −9.562667246049075, −9.083333378271487, −8.487915338211892, −7.787962821200370, −7.679064732256772, −7.050939778454609, −6.250581717803273, −5.799171839089829, −5.422350835741028, −4.542014999729784, −3.907349667642782, −3.267207753949366, −2.497426031401755, −2.167782155189871, −1.150382966321232, −0.4001995270347800, 0.4001995270347800, 1.150382966321232, 2.167782155189871, 2.497426031401755, 3.267207753949366, 3.907349667642782, 4.542014999729784, 5.422350835741028, 5.799171839089829, 6.250581717803273, 7.050939778454609, 7.679064732256772, 7.787962821200370, 8.487915338211892, 9.083333378271487, 9.562667246049075, 10.07172748423433, 10.55477562750277, 10.92789742735676, 11.45678054506663, 12.19094019654634, 12.63745436258497, 12.94229889181452, 13.49260754554848, 14.37065047710329

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