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Properties

Label 2-72450-1.1-c1-0-12
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 − 14-s + 16-s − 2·19-s − 23-s − 28-s + 2·29-s − 8·31-s + 32-s + 2·37-s − 2·38-s + 6·41-s − 4·43-s − 46-s − 8·47-s + 49-s − 12·53-s − 56-s + 2·58-s + 6·59-s − 6·61-s − 8·62-s + 64-s + 4·67-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.267·14-s + 1/4·16-s − 0.458·19-s − 0.208·23-s − 0.188·28-s + 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.328·37-s − 0.324·38-s + 0.937·41-s − 0.609·43-s − 0.147·46-s − 1.16·47-s + 1/7·49-s − 1.64·53-s − 0.133·56-s + 0.262·58-s + 0.781·59-s − 0.768·61-s − 1.01·62-s + 1/8·64-s + 0.488·67-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

\(L(1)\) \(\approx\) \(2.705902231\)
\(L(\frac12)\) \(\approx\) \(2.705902231\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.21300855401603, −13.51548217590709, −13.06733063111119, −12.69809671340151, −12.25152232788163, −11.64519242944949, −11.10247553462024, −10.76598642899080, −10.10483420954948, −9.567544290414684, −9.128528506050655, −8.384595985646725, −7.911363428664247, −7.332359493114736, −6.705664516159752, −6.319318773476299, −5.716108986062719, −5.179217504871208, −4.541558877854721, −4.024974297706345, −3.361966021737997, −2.890659666098169, −2.083576386463305, −1.527187486004052, −0.4629212710662993, 0.4629212710662993, 1.527187486004052, 2.083576386463305, 2.890659666098169, 3.361966021737997, 4.024974297706345, 4.541558877854721, 5.179217504871208, 5.716108986062719, 6.319318773476299, 6.705664516159752, 7.332359493114736, 7.911363428664247, 8.384595985646725, 9.128528506050655, 9.567544290414684, 10.10483420954948, 10.76598642899080, 11.10247553462024, 11.64519242944949, 12.25152232788163, 12.69809671340151, 13.06733063111119, 13.51548217590709, 14.21300855401603

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