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Properties

Label 2-72450-1.1-c1-0-24
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 + 3·13-s + 14-s + 16-s − 17-s − 4·22-s − 23-s + 3·26-s + 28-s − 4·31-s + 32-s − 34-s − 11·37-s + 10·41-s + 2·43-s − 4·44-s − 46-s + 11·47-s + 49-s + 3·52-s + 53-s + 56-s + 8·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.20·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.852·22-s − 0.208·23-s + 0.588·26-s + 0.188·28-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 1.80·37-s + 1.56·41-s + 0.304·43-s − 0.603·44-s − 0.147·46-s + 1.60·47-s + 1/7·49-s + 0.416·52-s + 0.137·53-s + 0.133·56-s + 1.04·59-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

\(L(1)\) \(\approx\) \(3.538226438\)
\(L(\frac12)\) \(\approx\) \(3.538226438\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.06350729202308, −13.63754113160421, −13.09878848344575, −12.69422403548239, −12.22543302871399, −11.63058643356796, −11.03361462360304, −10.71132448979420, −10.32611651981092, −9.601886357444167, −8.860220841160610, −8.530074876919196, −7.825464756269506, −7.389849948091674, −6.895986607003643, −6.139675696008374, −5.559172885886969, −5.353107668746327, −4.515159285916474, −4.063155134529414, −3.425790037887287, −2.737328552088964, −2.172665118454339, −1.480241666803881, −0.5395812614827990, 0.5395812614827990, 1.480241666803881, 2.172665118454339, 2.737328552088964, 3.425790037887287, 4.063155134529414, 4.515159285916474, 5.353107668746327, 5.559172885886969, 6.139675696008374, 6.895986607003643, 7.389849948091674, 7.825464756269506, 8.530074876919196, 8.860220841160610, 9.601886357444167, 10.32611651981092, 10.71132448979420, 11.03361462360304, 11.63058643356796, 12.22543302871399, 12.69422403548239, 13.09878848344575, 13.63754113160421, 14.06350729202308

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