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Properties

Label 2-72450-1.1-c1-0-85
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 + 4·13-s + 14-s + 16-s + 2·17-s + 2·19-s − 4·22-s − 23-s − 4·26-s − 28-s − 4·29-s − 8·31-s − 32-s − 2·34-s − 10·37-s − 2·38-s − 8·43-s + 4·44-s + 46-s − 8·47-s + 49-s + 4·52-s + 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.20·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.458·19-s − 0.852·22-s − 0.208·23-s − 0.784·26-s − 0.188·28-s − 0.742·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 1.64·37-s − 0.324·38-s − 1.21·43-s + 0.603·44-s + 0.147·46-s − 1.16·47-s + 1/7·49-s + 0.554·52-s + 0.274·53-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.36084392477267, −13.95375832338188, −13.23001841880068, −12.94338296816227, −12.18968784502236, −11.67683980667364, −11.47561291574226, −10.72026434060774, −10.32367947971651, −9.745233353474356, −9.157998163327123, −8.926444862930796, −8.313952089714685, −7.752194642664475, −7.086167390045575, −6.697343201083522, −6.173418807265104, −5.565465082180405, −5.046640472471310, −4.037669602610978, −3.430233535868117, −3.358543642568796, −2.074139306084910, −1.619146688346478, −0.9433101267693720, 0, 0.9433101267693720, 1.619146688346478, 2.074139306084910, 3.358543642568796, 3.430233535868117, 4.037669602610978, 5.046640472471310, 5.565465082180405, 6.173418807265104, 6.697343201083522, 7.086167390045575, 7.752194642664475, 8.313952089714685, 8.926444862930796, 9.157998163327123, 9.745233353474356, 10.32367947971651, 10.72026434060774, 11.47561291574226, 11.67683980667364, 12.18968784502236, 12.94338296816227, 13.23001841880068, 13.95375832338188, 14.36084392477267

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