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Properties

Label 2-1920-120.29-c0-0-4
Degree $2$
Conductor $1920$
Sign $1$
Analytic cond. $0.958204$
Root an. cond. $0.978879$
Motivic weight $0$
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 + 9-s − 15-s + 2·23-s + 25-s + 27-s + 2·29-s − 2·43-s − 45-s − 2·47-s + 49-s − 2·67-s + 2·69-s + 75-s + 81-s + 2·87-s − 2·101-s − 2·115-s + ⋯
L(s)  = 1  + 3-s − 5-s + 9-s − 15-s + 2·23-s + 25-s + 27-s + 2·29-s − 2·43-s − 45-s − 2·47-s + 49-s − 2·67-s + 2·69-s + 75-s + 81-s + 2·87-s − 2·101-s − 2·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1920} (449, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.454658268\)
\(L(\frac12)\) \(\approx\) \(1.454658268\)
\(L(1)\) 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 \( 1 + T \)
good7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )^{2} \)
29 \( ( 1 - T )^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 + T )^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 + T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.179210280473523518106314995696, −8.510962056244315159026050849397, −8.020613959539927655761921301808, −7.07599363653659849852651018625, −6.62185779670699238639830397820, −5.02238501407956788899052562966, −4.44338872261501732133845493065, −3.34354572797465442821691930728, −2.82767799087238506487837947276, −1.28822101635495953522537599373, 1.28822101635495953522537599373, 2.82767799087238506487837947276, 3.34354572797465442821691930728, 4.44338872261501732133845493065, 5.02238501407956788899052562966, 6.62185779670699238639830397820, 7.07599363653659849852651018625, 8.020613959539927655761921301808, 8.510962056244315159026050849397, 9.179210280473523518106314995696

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