Properties

Degree $2$
Conductor $72450$
Sign $-1$
Motivic weight $1$
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 + 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 − 8·47-s + 49-s + 2·52-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 − 1.16·47-s + 1/7·49-s + 0.277·52-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{72450} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.12117741582448, −13.92503543100214, −13.47567347955242, −12.74298396279184, −12.52925294408867, −11.89009406701758, −11.29562098756632, −11.09774998829028, −10.46834145504364, −9.744482160766805, −9.268198765280542, −8.832092038588604, −8.301294094077563, −7.385396832228009, −7.118561835173480, −6.447644992714732, −6.100319358881915, −5.499138613113521, −4.794264442945736, −4.235954944323767, −3.698824204073866, −3.249012962505881, −2.441393216910755, −1.745189053376929, −1.101733600105433, 0, 1.101733600105433, 1.745189053376929, 2.441393216910755, 3.249012962505881, 3.698824204073866, 4.235954944323767, 4.794264442945736, 5.499138613113521, 6.100319358881915, 6.447644992714732, 7.118561835173480, 7.385396832228009, 8.301294094077563, 8.832092038588604, 9.268198765280542, 9.744482160766805, 10.46834145504364, 11.09774998829028, 11.29562098756632, 11.89009406701758, 12.52925294408867, 12.74298396279184, 13.47567347955242, 13.92503543100214, 14.12117741582448

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