Properties

Label 2-100800-1.1-c1-0-255
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 4·11-s − 2·13-s + 2·17-s + 4·19-s − 10·29-s + 6·37-s + 6·41-s − 4·43-s − 8·47-s + 49-s − 6·53-s − 4·59-s + 10·61-s + 4·67-s + 16·71-s + 14·73-s − 4·77-s + 8·79-s + 4·83-s − 10·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.85·29-s + 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.520·59-s + 1.28·61-s + 0.488·67-s + 1.89·71-s + 1.63·73-s − 0.455·77-s + 0.900·79-s + 0.439·83-s − 1.05·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 100800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.98185922219106, −13.48831241596662, −12.93475291750475, −12.65437929234719, −12.05459786401625, −11.47362467164189, −10.99727660563270, −10.74789481596552, −9.869006323117043, −9.588699356498321, −9.275925174928096, −8.274748200400645, −7.830558462511191, −7.786999477221252, −6.955950457147894, −6.494109161548620, −5.565523163240397, −5.357341085903523, −4.930445219616171, −4.116947295083129, −3.562538282184670, −2.878101846015721, −2.338427303907269, −1.680642829502003, −0.8323018351488837, 0, 0.8323018351488837, 1.680642829502003, 2.338427303907269, 2.878101846015721, 3.562538282184670, 4.116947295083129, 4.930445219616171, 5.357341085903523, 5.565523163240397, 6.494109161548620, 6.955950457147894, 7.786999477221252, 7.830558462511191, 8.274748200400645, 9.275925174928096, 9.588699356498321, 9.869006323117043, 10.74789481596552, 10.99727660563270, 11.47362467164189, 12.05459786401625, 12.65437929234719, 12.93475291750475, 13.48831241596662, 13.98185922219106

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