Properties

Label 2-72450-1.1-c1-0-26
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 − 6·11-s − 5·13-s + 14-s + 16-s − 6·17-s + 2·19-s + 6·22-s − 23-s + 5·26-s − 28-s − 9·29-s − 4·31-s − 32-s + 6·34-s + 37-s − 2·38-s + 9·41-s − 11·43-s − 6·44-s + 46-s − 3·47-s + 49-s − 5·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.80·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s + 1.27·22-s − 0.208·23-s + 0.980·26-s − 0.188·28-s − 1.67·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.164·37-s − 0.324·38-s + 1.40·41-s − 1.67·43-s − 0.904·44-s + 0.147·46-s − 0.437·47-s + 1/7·49-s − 0.693·52-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−14.64670380546218, −13.66040623823388, −13.26425319559826, −12.84661386249169, −12.49206319892002, −11.58788725940548, −11.39740733501358, −10.67583568469404, −10.33930359010228, −9.671217805734267, −9.495983692784593, −8.727761348239895, −8.257940784466335, −7.632964805791227, −7.201978390888928, −6.920014571007532, −5.971437804663539, −5.514474426117709, −4.983948835788283, −4.362217286340381, −3.515876331819860, −2.839751066004065, −2.236888951899006, −1.929619697406189, −0.5479065826440247, 0, 0.5479065826440247, 1.929619697406189, 2.236888951899006, 2.839751066004065, 3.515876331819860, 4.362217286340381, 4.983948835788283, 5.514474426117709, 5.971437804663539, 6.920014571007532, 7.201978390888928, 7.632964805791227, 8.257940784466335, 8.727761348239895, 9.495983692784593, 9.671217805734267, 10.33930359010228, 10.67583568469404, 11.39740733501358, 11.58788725940548, 12.49206319892002, 12.84661386249169, 13.26425319559826, 13.66040623823388, 14.64670380546218

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