Properties

Label 2-72450-1.1-c1-0-106
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 − 6·13-s − 14-s + 16-s + 6·19-s + 4·22-s − 23-s − 6·26-s − 28-s − 8·29-s + 8·31-s + 32-s − 2·37-s + 6·38-s + 2·41-s − 8·43-s + 4·44-s − 46-s + 12·47-s + 49-s − 6·52-s − 2·53-s − 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.37·19-s + 0.852·22-s − 0.208·23-s − 1.17·26-s − 0.188·28-s − 1.48·29-s + 1.43·31-s + 0.176·32-s − 0.328·37-s + 0.973·38-s + 0.312·41-s − 1.21·43-s + 0.603·44-s − 0.147·46-s + 1.75·47-s + 1/7·49-s − 0.832·52-s − 0.274·53-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: \(1\)
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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.09838980504376, −14.02750652070822, −13.46704010136012, −12.80090543396725, −12.25474853937219, −11.91774671478528, −11.67682377392597, −10.91453373645173, −10.32457086218236, −9.749985312455648, −9.367757296753802, −8.981636656630915, −7.946724060940014, −7.644598697909222, −6.970696873854962, −6.683108646722118, −5.923453277061285, −5.457324774535251, −4.843750859161138, −4.313208837085370, −3.702422298325287, −3.092592771195979, −2.547223913501232, −1.772645183702904, −1.046812029357416, 0, 1.046812029357416, 1.772645183702904, 2.547223913501232, 3.092592771195979, 3.702422298325287, 4.313208837085370, 4.843750859161138, 5.457324774535251, 5.923453277061285, 6.683108646722118, 6.970696873854962, 7.644598697909222, 7.946724060940014, 8.981636656630915, 9.367757296753802, 9.749985312455648, 10.32457086218236, 10.91453373645173, 11.67682377392597, 11.91774671478528, 12.25474853937219, 12.80090543396725, 13.46704010136012, 14.02750652070822, 14.09838980504376

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