Properties

Label 2-4800-1.1-c1-0-30
Degree $2$
Conductor $4800$
Sign $1$
Analytic cond. $38.3281$
Root an. cond. $6.19097$
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  − 3-s − 7-s + 9-s + 6·11-s + 5·13-s + 6·17-s + 5·19-s + 21-s − 6·23-s − 27-s + 6·29-s + 31-s − 6·33-s + 2·37-s − 5·39-s + 43-s + 6·47-s − 6·49-s − 6·51-s − 12·53-s − 5·57-s − 6·59-s + 13·61-s − 63-s − 11·67-s + 6·69-s − 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.80·11-s + 1.38·13-s + 1.45·17-s + 1.14·19-s + 0.218·21-s − 1.25·23-s − 0.192·27-s + 1.11·29-s + 0.179·31-s − 1.04·33-s + 0.328·37-s − 0.800·39-s + 0.152·43-s + 0.875·47-s − 6/7·49-s − 0.840·51-s − 1.64·53-s − 0.662·57-s − 0.781·59-s + 1.66·61-s − 0.125·63-s − 1.34·67-s + 0.722·69-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(38.3281\)
Root analytic conductor: \(6.19097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.098755874\)
\(L(\frac12)\) \(\approx\) \(2.098755874\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−8.243597547750200954321639537018, −7.54038315127717275337161988087, −6.58837472702591983831965532194, −6.16571856132998816140053150198, −5.57958254615529070693398763832, −4.48317997032114878344902590623, −3.73650586061389001500010789061, −3.14969814812418521417534836385, −1.52862254168278113235246302632, −0.942550376347668032286962617546, 0.942550376347668032286962617546, 1.52862254168278113235246302632, 3.14969814812418521417534836385, 3.73650586061389001500010789061, 4.48317997032114878344902590623, 5.57958254615529070693398763832, 6.16571856132998816140053150198, 6.58837472702591983831965532194, 7.54038315127717275337161988087, 8.243597547750200954321639537018

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