Properties

Label 2-72450-1.1-c1-0-101
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 − 2·11-s + 6·13-s − 14-s + 16-s − 6·17-s − 2·22-s + 23-s + 6·26-s − 28-s + 4·29-s − 2·31-s + 32-s − 6·34-s − 4·37-s + 2·41-s + 4·43-s − 2·44-s + 46-s + 49-s + 6·52-s + 6·53-s − 56-s + 4·58-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 − 1.45·17-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 − 1.02·34-s − 0.657·37-s + 0.312·41-s + 0.609·43-s − 0.301·44-s + 0.147·46-s + 1/7·49-s + 0.832·52-s + 0.824·53-s − 0.133·56-s + 0.525·58-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: $\chi_{72450} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−14.15186229328985, −13.75821866695076, −13.37149536520961, −12.92913821330352, −12.58371669647273, −11.81800737924831, −11.35963378393646, −10.91961139711363, −10.45585883422021, −10.02359140106335, −9.088495710424194, −8.766236883452666, −8.336966499157762, −7.510725627229990, −7.076203598878625, −6.379822425157138, −6.117288612829042, −5.468403759365634, −4.873399914590519, −4.157439293998762, −3.840340879129332, −3.031180919426583, −2.572874712606028, −1.782949928213085, −1.031096301372602, 0, 1.031096301372602, 1.782949928213085, 2.572874712606028, 3.031180919426583, 3.840340879129332, 4.157439293998762, 4.873399914590519, 5.468403759365634, 6.117288612829042, 6.379822425157138, 7.076203598878625, 7.510725627229990, 8.336966499157762, 8.766236883452666, 9.088495710424194, 10.02359140106335, 10.45585883422021, 10.91961139711363, 11.35963378393646, 11.81800737924831, 12.58371669647273, 12.92913821330352, 13.37149536520961, 13.75821866695076, 14.15186229328985

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