Properties

Label 2-100800-1.1-c1-0-19
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

\(L(1)\) \(\approx\) \(0.8394516501\)
\(L(\frac12)\) \(\approx\) \(0.8394516501\)
\(L(\frac{3}{2})\) 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

−13.83275145919606, −13.38395931841878, −12.50769661130856, −12.18539753242353, −12.03791630862153, −11.47746439984465, −10.61395803904100, −10.24950931261517, −9.818529550511175, −9.459442026590839, −8.647305352610609, −8.378660515515713, −7.711306580641242, −7.215720063217799, −6.611911025188266, −6.120823480457950, −5.754782568079645, −4.949060100825217, −4.405198028640253, −3.839575646220533, −3.315041468689279, −2.702894299648950, −1.728111086616192, −1.514023488467416, −0.2757708946431827, 0.2757708946431827, 1.514023488467416, 1.728111086616192, 2.702894299648950, 3.315041468689279, 3.839575646220533, 4.405198028640253, 4.949060100825217, 5.754782568079645, 6.120823480457950, 6.611911025188266, 7.215720063217799, 7.711306580641242, 8.378660515515713, 8.647305352610609, 9.459442026590839, 9.818529550511175, 10.24950931261517, 10.61395803904100, 11.47746439984465, 12.03791630862153, 12.18539753242353, 12.50769661130856, 13.38395931841878, 13.83275145919606

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