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Properties

Label 2-72450-1.1-c1-0-8
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 $0$

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 − 4·19-s − 4·22-s − 23-s − 2·26-s − 28-s − 8·29-s − 8·31-s − 32-s − 6·37-s + 4·38-s + 8·41-s + 8·43-s + 4·44-s + 46-s + 49-s + 2·52-s − 12·53-s + 56-s + 8·58-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 − 0.917·19-s − 0.852·22-s − 0.208·23-s − 0.392·26-s − 0.188·28-s − 1.48·29-s − 1.43·31-s − 0.176·32-s − 0.986·37-s + 0.648·38-s + 1.24·41-s + 1.21·43-s + 0.603·44-s + 0.147·46-s + 1/7·49-s + 0.277·52-s − 1.64·53-s + 0.133·56-s + 1.05·58-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: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.025518689\)
\(L(\frac12)\) \(\approx\) \(1.025518689\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−14.34540119895646, −13.51318480220195, −13.05598886726707, −12.51625201995237, −12.14978000156439, −11.38296120107348, −11.06448347376599, −10.67052007369751, −9.981501666139077, −9.328827927934328, −9.165406517745537, −8.625933830133462, −8.039663767281326, −7.315679398817847, −7.027675244197326, −6.329944107913640, −5.897995779342274, −5.419460723437826, −4.363218606008361, −3.917239008866300, −3.430742053102753, −2.593244034345353, −1.823450541712309, −1.376302535451347, −0.3786628390406170, 0.3786628390406170, 1.376302535451347, 1.823450541712309, 2.593244034345353, 3.430742053102753, 3.917239008866300, 4.363218606008361, 5.419460723437826, 5.897995779342274, 6.329944107913640, 7.027675244197326, 7.315679398817847, 8.039663767281326, 8.625933830133462, 9.165406517745537, 9.328827927934328, 9.981501666139077, 10.67052007369751, 11.06448347376599, 11.38296120107348, 12.14978000156439, 12.51625201995237, 13.05598886726707, 13.51318480220195, 14.34540119895646

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