Properties

Label 2-73920-1.1-c1-0-139
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 7-s + 9-s + 11-s − 2·13-s + 15-s + 6·17-s + 4·19-s − 21-s + 25-s − 27-s − 6·29-s − 4·31-s − 33-s − 35-s − 2·37-s + 2·39-s + 6·41-s + 4·43-s − 45-s + 49-s − 6·51-s − 6·53-s − 55-s − 4·57-s − 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s − 0.169·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.134·55-s − 0.529·57-s − 0.256·61-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 73920,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−14.41881726875175, −13.99213551794461, −13.23929878097605, −12.61213225631413, −12.35946449603037, −11.81971159707486, −11.32230908407345, −11.01140488701910, −10.29641939590899, −9.853803247098583, −9.289487916568890, −8.881108404982928, −7.909905961899235, −7.675142673146161, −7.291104534031596, −6.586831257072605, −5.905284703185330, −5.373748326795779, −5.052928673992675, −4.266216024985443, −3.707081485056012, −3.176446100002806, −2.340363323201458, −1.493304583115605, −0.9233429227471460, 0, 0.9233429227471460, 1.493304583115605, 2.340363323201458, 3.176446100002806, 3.707081485056012, 4.266216024985443, 5.052928673992675, 5.373748326795779, 5.905284703185330, 6.586831257072605, 7.291104534031596, 7.675142673146161, 7.909905961899235, 8.881108404982928, 9.289487916568890, 9.853803247098583, 10.29641939590899, 11.01140488701910, 11.32230908407345, 11.81971159707486, 12.35946449603037, 12.61213225631413, 13.23929878097605, 13.99213551794461, 14.41881726875175

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