Properties

Label 2-10920-1.1-c1-0-1
Degree $2$
Conductor $10920$
Sign $1$
Analytic cond. $87.1966$
Root an. cond. $9.33791$
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 + 2·11-s + 13-s + 15-s − 2·17-s − 6·19-s − 21-s − 4·23-s + 25-s − 27-s − 8·29-s + 8·31-s − 2·33-s − 35-s + 2·37-s − 39-s + 6·41-s − 4·43-s − 45-s + 4·47-s + 49-s + 2·51-s + 6·53-s − 2·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.258·15-s − 0.485·17-s − 1.37·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.348·33-s − 0.169·35-s + 0.328·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10920\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(87.1966\)
Root analytic conductor: \(9.33791\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.261781298\)
\(L(\frac12)\) \(\approx\) \(1.261781298\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 - T \)
good11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.71065944591898, −15.90345722191262, −15.40054158903784, −14.88351217859291, −14.34165779864409, −13.54282142914251, −13.09017280420197, −12.33034994090546, −11.87307881913682, −11.34427668754575, −10.74877843960709, −10.31102872701643, −9.396989406529852, −8.875229380617744, −8.144582273313841, −7.621305739193739, −6.797522874975448, −6.236024917945326, −5.700675189350217, −4.673735766422291, −4.253480122419841, −3.608478220744390, −2.429611489194497, −1.643198372611369, −0.5431821408183072, 0.5431821408183072, 1.643198372611369, 2.429611489194497, 3.608478220744390, 4.253480122419841, 4.673735766422291, 5.700675189350217, 6.236024917945326, 6.797522874975448, 7.621305739193739, 8.144582273313841, 8.875229380617744, 9.396989406529852, 10.31102872701643, 10.74877843960709, 11.34427668754575, 11.87307881913682, 12.33034994090546, 13.09017280420197, 13.54282142914251, 14.34165779864409, 14.88351217859291, 15.40054158903784, 15.90345722191262, 16.71065944591898

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