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Properties

Label 2-72450-1.1-c1-0-126
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s + 5·11-s + 14-s + 16-s + 2·17-s − 3·19-s + 5·22-s − 23-s + 28-s + 2·31-s + 32-s + 2·34-s − 2·37-s − 3·38-s − 5·41-s − 10·43-s + 5·44-s − 46-s − 7·47-s + 49-s − 11·53-s + 56-s − 4·59-s − 2·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.50·11-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.688·19-s + 1.06·22-s − 0.208·23-s + 0.188·28-s + 0.359·31-s + 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.486·38-s − 0.780·41-s − 1.52·43-s + 0.753·44-s − 0.147·46-s − 1.02·47-s + 1/7·49-s − 1.51·53-s + 0.133·56-s − 0.520·59-s − 0.256·61-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: \(1\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−14.27874179466370, −13.88277206104726, −13.54015696810662, −12.72638657833928, −12.39054081838783, −11.92186505195504, −11.39616545113869, −11.08225424653784, −10.36278446837548, −9.855299720830406, −9.320656289574068, −8.740614124341421, −8.076823859134815, −7.807336418803602, −6.759792929494829, −6.635795611644795, −6.153069060052012, −5.281610412049059, −4.923931023571059, −4.262731490901563, −3.683409389796471, −3.280111115589512, −2.400045669352061, −1.634452327708298, −1.248561246974353, 0, 1.248561246974353, 1.634452327708298, 2.400045669352061, 3.280111115589512, 3.683409389796471, 4.262731490901563, 4.923931023571059, 5.281610412049059, 6.153069060052012, 6.635795611644795, 6.759792929494829, 7.807336418803602, 8.076823859134815, 8.740614124341421, 9.320656289574068, 9.855299720830406, 10.36278446837548, 11.08225424653784, 11.39616545113869, 11.92186505195504, 12.39054081838783, 12.72638657833928, 13.54015696810662, 13.88277206104726, 14.27874179466370

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