Properties

Label 2-72450-1.1-c1-0-2
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 + 4·22-s − 23-s + 3·26-s − 28-s − 29-s − 2·31-s − 32-s + 5·37-s − 5·41-s + 7·43-s − 4·44-s + 46-s − 3·47-s + 49-s − 3·52-s + 12·53-s + 56-s + 58-s + 2·59-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 + 0.852·22-s − 0.208·23-s + 0.588·26-s − 0.188·28-s − 0.185·29-s − 0.359·31-s − 0.176·32-s + 0.821·37-s − 0.780·41-s + 1.06·43-s − 0.603·44-s + 0.147·46-s − 0.437·47-s + 1/7·49-s − 0.416·52-s + 1.64·53-s + 0.133·56-s + 0.131·58-s + 0.260·59-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: $\chi_{72450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5596285527\)
\(L(\frac12)\) \(\approx\) \(0.5596285527\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−14.08318256605601, −13.56115416575589, −13.03309461856972, −12.55686842220901, −12.13902752928070, −11.51430748140104, −10.97794297491696, −10.50181411447535, −10.00676819792430, −9.629030828898277, −9.091504510908469, −8.436573147670028, −7.991673775947078, −7.452042556619309, −7.040603901729900, −6.441919636299726, −5.657239568689145, −5.387728981779124, −4.607581329914660, −3.950905921663316, −3.162443633722176, −2.553742542292472, −2.162510150614954, −1.187770515782669, −0.2896875594691245, 0.2896875594691245, 1.187770515782669, 2.162510150614954, 2.553742542292472, 3.162443633722176, 3.950905921663316, 4.607581329914660, 5.387728981779124, 5.657239568689145, 6.441919636299726, 7.040603901729900, 7.452042556619309, 7.991673775947078, 8.436573147670028, 9.091504510908469, 9.629030828898277, 10.00676819792430, 10.50181411447535, 10.97794297491696, 11.51430748140104, 12.13902752928070, 12.55686842220901, 13.03309461856972, 13.56115416575589, 14.08318256605601

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