Properties

Label 2-73920-1.1-c1-0-33
Degree $2$
Conductor $73920$
Sign $1$
Analytic cond. $590.254$
Root an. cond. $24.2951$
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 − 5-s − 7-s + 9-s + 11-s + 6·13-s − 15-s − 7·17-s − 5·19-s − 21-s + 23-s + 25-s + 27-s + 5·29-s + 8·31-s + 33-s + 35-s + 2·37-s + 6·39-s + 12·41-s − 11·43-s − 45-s − 8·47-s + 49-s − 7·51-s + 11·53-s − 55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s − 0.258·15-s − 1.69·17-s − 1.14·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.928·29-s + 1.43·31-s + 0.174·33-s + 0.169·35-s + 0.328·37-s + 0.960·39-s + 1.87·41-s − 1.67·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.980·51-s + 1.51·53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73920\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(590.254\)
Root analytic conductor: \(24.2951\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{73920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 73920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.574200945\)
\(L(\frac12)\) \(\approx\) \(2.574200945\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−13.89897895285660, −13.54208489012480, −13.18291099878485, −12.78657710710460, −12.00884497105837, −11.63409995007930, −10.93248938244291, −10.70041484697667, −10.09916517407432, −9.363357062048350, −8.872971121836718, −8.515350124834886, −8.166019555311001, −7.451841852084630, −6.663611587831652, −6.394282385515099, −6.065385031442509, −4.932721825326802, −4.377711015943798, −4.055892039982798, −3.350526301028488, −2.769818099556902, −2.109012942409227, −1.317452386631707, −0.5303162762638641, 0.5303162762638641, 1.317452386631707, 2.109012942409227, 2.769818099556902, 3.350526301028488, 4.055892039982798, 4.377711015943798, 4.932721825326802, 6.065385031442509, 6.394282385515099, 6.663611587831652, 7.451841852084630, 8.166019555311001, 8.515350124834886, 8.872971121836718, 9.363357062048350, 10.09916517407432, 10.70041484697667, 10.93248938244291, 11.63409995007930, 12.00884497105837, 12.78657710710460, 13.18291099878485, 13.54208489012480, 13.89897895285660

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