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Properties

Label 2-72450-1.1-c1-0-31
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 + 2·11-s + 6·13-s + 14-s + 16-s + 6·19-s − 2·22-s − 23-s − 6·26-s − 28-s − 4·29-s − 2·31-s − 32-s + 2·37-s − 6·38-s − 2·41-s + 10·43-s + 2·44-s + 46-s + 6·47-s + 49-s + 6·52-s + 6·53-s + 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.603·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.37·19-s − 0.426·22-s − 0.208·23-s − 1.17·26-s − 0.188·28-s − 0.742·29-s − 0.359·31-s − 0.176·32-s + 0.328·37-s − 0.973·38-s − 0.312·41-s + 1.52·43-s + 0.301·44-s + 0.147·46-s + 0.875·47-s + 1/7·49-s + 0.832·52-s + 0.824·53-s + 0.133·56-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

\(L(1)\) \(\approx\) \(2.079100221\)
\(L(\frac12)\) \(\approx\) \(2.079100221\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.06641102298232, −13.68211196487091, −13.02962811017850, −12.67077617966477, −11.82112184169468, −11.62956693932185, −11.07764675968029, −10.50853459975844, −10.10989155110766, −9.371035709187750, −9.010541545689126, −8.743978934344116, −7.902012682010506, −7.476182330541472, −7.007048184710087, −6.255440661990025, −5.884645668357995, −5.459416171454707, −4.432594218838466, −3.858983265628414, −3.337260957570018, −2.741928050808054, −1.802815858181732, −1.224597421532060, −0.5956161871072987, 0.5956161871072987, 1.224597421532060, 1.802815858181732, 2.741928050808054, 3.337260957570018, 3.858983265628414, 4.432594218838466, 5.459416171454707, 5.884645668357995, 6.255440661990025, 7.007048184710087, 7.476182330541472, 7.902012682010506, 8.743978934344116, 9.010541545689126, 9.371035709187750, 10.10989155110766, 10.50853459975844, 11.07764675968029, 11.62956693932185, 11.82112184169468, 12.67077617966477, 13.02962811017850, 13.68211196487091, 14.06641102298232

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