Properties

Label 2-100800-1.1-c1-0-10
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 − 6·11-s − 13-s + 3·17-s − 4·19-s + 3·23-s + 3·29-s − 5·31-s − 10·37-s − 9·41-s + 43-s + 49-s − 9·53-s − 9·59-s − 11·61-s + 4·67-s − 12·71-s + 10·73-s − 6·77-s + 10·79-s + 9·83-s + 6·89-s − 91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.80·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s + 0.625·23-s + 0.557·29-s − 0.898·31-s − 1.64·37-s − 1.40·41-s + 0.152·43-s + 1/7·49-s − 1.23·53-s − 1.17·59-s − 1.40·61-s + 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.683·77-s + 1.12·79-s + 0.987·83-s + 0.635·89-s − 0.104·91-s − 1.42·97-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

\(L(1)\) \(\approx\) \(0.6019414394\)
\(L(\frac12)\) \(\approx\) \(0.6019414394\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
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

−13.77447145511060, −13.25131504462570, −12.73528927964535, −12.30299074168303, −11.98274130194934, −11.01649405518251, −10.87103120325348, −10.34724775935695, −9.973921092616036, −9.240553416773350, −8.714450138068026, −8.229810369342053, −7.691138493495677, −7.407169714452295, −6.666628217026224, −6.128948585077189, −5.427852018741320, −4.902261956407778, −4.809999229602979, −3.735573219679289, −3.230275290884830, −2.633983432167070, −1.978243794037038, −1.366754213165334, −0.2351288145553258, 0.2351288145553258, 1.366754213165334, 1.978243794037038, 2.633983432167070, 3.230275290884830, 3.735573219679289, 4.809999229602979, 4.902261956407778, 5.427852018741320, 6.128948585077189, 6.666628217026224, 7.407169714452295, 7.691138493495677, 8.229810369342053, 8.714450138068026, 9.240553416773350, 9.973921092616036, 10.34724775935695, 10.87103120325348, 11.01649405518251, 11.98274130194934, 12.30299074168303, 12.73528927964535, 13.25131504462570, 13.77447145511060

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