Properties

Label 2-100800-1.1-c1-0-197
Degree $2$
Conductor $100800$
Sign $1$
Analytic cond. $804.892$
Root an. cond. $28.3706$
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  − 7-s + 4·11-s + 3·13-s + 7·17-s + 6·19-s + 9·23-s + 3·29-s − 7·31-s + 10·37-s + 41-s − 13·43-s − 2·47-s + 49-s + 53-s − 11·59-s − 13·61-s − 8·71-s + 8·73-s − 4·77-s + 4·79-s − 7·83-s + 14·89-s − 3·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.20·11-s + 0.832·13-s + 1.69·17-s + 1.37·19-s + 1.87·23-s + 0.557·29-s − 1.25·31-s + 1.64·37-s + 0.156·41-s − 1.98·43-s − 0.291·47-s + 1/7·49-s + 0.137·53-s − 1.43·59-s − 1.66·61-s − 0.949·71-s + 0.936·73-s − 0.455·77-s + 0.450·79-s − 0.768·83-s + 1.48·89-s − 0.314·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100800\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(804.892\)
Root analytic conductor: \(28.3706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{100800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 100800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.901418712\)
\(L(\frac12)\) \(\approx\) \(3.901418712\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.66643732339124, −13.36643869930741, −12.74141845358576, −12.23763335637450, −11.83090111479301, −11.30011086342004, −10.92484846919689, −10.25352642152042, −9.652826457795699, −9.360517463663003, −8.904721368360482, −8.299182881437399, −7.556369164939299, −7.360045935677354, −6.570157780876113, −6.203206714419819, −5.601229212844268, −5.062652111535557, −4.475628514577794, −3.658228730906535, −3.193264705778436, −3.012263381915309, −1.702807736411274, −1.223420391174653, −0.7138016707083573, 0.7138016707083573, 1.223420391174653, 1.702807736411274, 3.012263381915309, 3.193264705778436, 3.658228730906535, 4.475628514577794, 5.062652111535557, 5.601229212844268, 6.203206714419819, 6.570157780876113, 7.360045935677354, 7.556369164939299, 8.299182881437399, 8.904721368360482, 9.360517463663003, 9.652826457795699, 10.25352642152042, 10.92484846919689, 11.30011086342004, 11.83090111479301, 12.23763335637450, 12.74141845358576, 13.36643869930741, 13.66643732339124

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