Properties

Label 2-58800-1.1-c1-0-27
Degree $2$
Conductor $58800$
Sign $1$
Analytic cond. $469.520$
Root an. cond. $21.6684$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 6·11-s − 2·13-s + 4·17-s − 6·19-s − 27-s − 2·29-s − 10·31-s − 6·33-s + 4·37-s + 2·39-s − 2·41-s − 4·43-s − 4·51-s − 6·53-s + 6·57-s − 8·59-s + 2·61-s − 16·67-s − 10·71-s − 6·73-s − 4·79-s + 81-s − 8·83-s + 2·87-s − 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s − 0.192·27-s − 0.371·29-s − 1.79·31-s − 1.04·33-s + 0.657·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s − 0.560·51-s − 0.824·53-s + 0.794·57-s − 1.04·59-s + 0.256·61-s − 1.95·67-s − 1.18·71-s − 0.702·73-s − 0.450·79-s + 1/9·81-s − 0.878·83-s + 0.214·87-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(469.520\)
Root analytic conductor: \(21.6684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{58800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.273308189\)
\(L(\frac12)\) \(\approx\) \(1.273308189\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.55720940926343, −13.88960050755261, −13.22673428978196, −12.68528107672228, −12.26311669023793, −11.87369283040604, −11.24017218563540, −10.94979004649968, −10.15982059992440, −9.823949777507460, −9.074944315102024, −8.889235168470486, −8.061825256622262, −7.374374078440252, −7.027702408746576, −6.318036100925590, −5.959657252582075, −5.377123893541459, −4.525553129581115, −4.209692678516328, −3.527060587013745, −2.872427364152544, −1.658303884875987, −1.594038122992191, −0.3984373913337498, 0.3984373913337498, 1.594038122992191, 1.658303884875987, 2.872427364152544, 3.527060587013745, 4.209692678516328, 4.525553129581115, 5.377123893541459, 5.959657252582075, 6.318036100925590, 7.027702408746576, 7.374374078440252, 8.061825256622262, 8.889235168470486, 9.074944315102024, 9.823949777507460, 10.15982059992440, 10.94979004649968, 11.24017218563540, 11.87369283040604, 12.26311669023793, 12.68528107672228, 13.22673428978196, 13.88960050755261, 14.55720940926343

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