Properties

Label 2-10920-1.1-c1-0-18
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 7-s + 9-s + 4·11-s − 13-s − 15-s − 2·17-s − 21-s + 4·23-s + 25-s + 27-s − 10·29-s − 4·31-s + 4·33-s + 35-s + 6·37-s − 39-s − 2·41-s − 4·43-s − 45-s + 49-s − 2·51-s − 14·53-s − 4·55-s + 4·59-s − 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s − 0.485·17-s − 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.160·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s − 1.92·53-s − 0.539·55-s + 0.520·59-s − 1.28·61-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: \(1\)
Selberg data: \((2,\ 10920,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
13 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.68944265368387, −16.35508338273716, −15.48042483682562, −15.02479534152539, −14.64704142402422, −14.02505589141623, −13.30296958840780, −12.84041492861318, −12.30773425260278, −11.43438653362622, −11.21526568438913, −10.36104240064980, −9.511565158281282, −9.203746240448025, −8.700267821057358, −7.769273980007437, −7.360822792245006, −6.632515146106294, −6.079053729775924, −5.080639174135736, −4.366500371161671, −3.659866362112842, −3.158784229839195, −2.130621580473238, −1.302642364193455, 0, 1.302642364193455, 2.130621580473238, 3.158784229839195, 3.659866362112842, 4.366500371161671, 5.080639174135736, 6.079053729775924, 6.632515146106294, 7.360822792245006, 7.769273980007437, 8.700267821057358, 9.203746240448025, 9.511565158281282, 10.36104240064980, 11.21526568438913, 11.43438653362622, 12.30773425260278, 12.84041492861318, 13.30296958840780, 14.02505589141623, 14.64704142402422, 15.02479534152539, 15.48042483682562, 16.35508338273716, 16.68944265368387

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