Properties

Label 2-72450-1.1-c1-0-129
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 + 2·11-s − 4·13-s + 14-s + 16-s − 6·17-s − 4·19-s − 2·22-s − 23-s + 4·26-s − 28-s − 2·29-s + 2·31-s − 32-s + 6·34-s − 4·37-s + 4·38-s + 2·41-s − 4·43-s + 2·44-s + 46-s − 4·47-s + 49-s − 4·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.603·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.426·22-s − 0.208·23-s + 0.784·26-s − 0.188·28-s − 0.371·29-s + 0.359·31-s − 0.176·32-s + 1.02·34-s − 0.657·37-s + 0.648·38-s + 0.312·41-s − 0.609·43-s + 0.301·44-s + 0.147·46-s − 0.583·47-s + 1/7·49-s − 0.554·52-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: \(2\)
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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−14.65991367203135, −14.15952508434856, −13.65054105291762, −12.88920514050584, −12.66168846291263, −12.03135737005351, −11.57734054493938, −10.86679630033598, −10.74921242688610, −9.832317118148992, −9.592657045190960, −9.133422966788908, −8.380495757662074, −8.234350662644722, −7.240982115920280, −7.021789079871049, −6.375250282674634, −6.036748192446847, −5.125888521204855, −4.543080076016032, −4.045411768748350, −3.204259738252459, −2.591531201931231, −1.968669818369002, −1.364115622226002, 0, 0, 1.364115622226002, 1.968669818369002, 2.591531201931231, 3.204259738252459, 4.045411768748350, 4.543080076016032, 5.125888521204855, 6.036748192446847, 6.375250282674634, 7.021789079871049, 7.240982115920280, 8.234350662644722, 8.380495757662074, 9.133422966788908, 9.592657045190960, 9.832317118148992, 10.74921242688610, 10.86679630033598, 11.57734054493938, 12.03135737005351, 12.66168846291263, 12.88920514050584, 13.65054105291762, 14.15952508434856, 14.65991367203135

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