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Properties

Label 2-72450-1.1-c1-0-38
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 − 4·11-s + 14-s + 16-s − 6·17-s + 4·22-s + 23-s − 28-s − 4·29-s − 4·31-s − 32-s + 6·34-s − 8·37-s + 4·41-s + 4·43-s − 4·44-s − 46-s + 12·47-s + 49-s − 10·53-s + 56-s + 4·58-s − 6·61-s + 4·62-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 + 0.852·22-s + 0.208·23-s − 0.188·28-s − 0.742·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s − 1.31·37-s + 0.624·41-s + 0.609·43-s − 0.603·44-s − 0.147·46-s + 1.75·47-s + 1/7·49-s − 1.37·53-s + 0.133·56-s + 0.525·58-s − 0.768·61-s + 0.508·62-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−14.29613997891177, −13.78911442782220, −13.36418687449176, −12.68600181735684, −12.50958317229162, −11.81164806483836, −11.01057558932929, −10.80523126793995, −10.52492837636251, −9.645779067176114, −9.289803970738921, −8.857478579267630, −8.259745774807106, −7.663302256100596, −7.269609657031826, −6.703285435250266, −6.111609317040497, −5.530413155417700, −4.971204269473177, −4.270030397174204, −3.567375916606256, −2.896608251872541, −2.258710623339700, −1.799610078570826, −0.6668179276670026, 0, 0.6668179276670026, 1.799610078570826, 2.258710623339700, 2.896608251872541, 3.567375916606256, 4.270030397174204, 4.971204269473177, 5.530413155417700, 6.111609317040497, 6.703285435250266, 7.269609657031826, 7.663302256100596, 8.259745774807106, 8.857478579267630, 9.289803970738921, 9.645779067176114, 10.52492837636251, 10.80523126793995, 11.01057558932929, 11.81164806483836, 12.50958317229162, 12.68600181735684, 13.36418687449176, 13.78911442782220, 14.29613997891177

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