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Properties

Label 2-72450-1.1-c1-0-128
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 4·11-s − 2·13-s − 14-s + 16-s − 6·17-s − 4·19-s + 4·22-s + 23-s + 2·26-s + 28-s − 2·29-s − 8·31-s − 32-s + 6·34-s − 6·37-s + 4·38-s + 6·41-s + 4·43-s − 4·44-s − 46-s + 49-s − 2·52-s − 6·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.20·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.852·22-s + 0.208·23-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.986·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s − 0.603·44-s − 0.147·46-s + 1/7·49-s − 0.277·52-s − 0.824·53-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: \(2\)
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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.78926562651963, −14.19827325102081, −13.43751762502104, −13.12265994245359, −12.57024241620902, −12.14296518712235, −11.38106029362348, −10.89643586570926, −10.69900022590421, −10.16158389469649, −9.454845166758981, −8.864517231673024, −8.723280914155462, −7.881817790036910, −7.474303178762655, −7.119871933083628, −6.320015371864409, −5.876811725718083, −5.118955734427528, −4.685664293577984, −4.020658933408011, −3.210304219983847, −2.405834336343953, −2.145719741431743, −1.338906085478091, 0, 0, 1.338906085478091, 2.145719741431743, 2.405834336343953, 3.210304219983847, 4.020658933408011, 4.685664293577984, 5.118955734427528, 5.876811725718083, 6.320015371864409, 7.119871933083628, 7.474303178762655, 7.881817790036910, 8.723280914155462, 8.864517231673024, 9.454845166758981, 10.16158389469649, 10.69900022590421, 10.89643586570926, 11.38106029362348, 12.14296518712235, 12.57024241620902, 13.12265994245359, 13.43751762502104, 14.19827325102081, 14.78926562651963

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