Properties

Label 2-6800-1.1-c1-0-59
Degree $2$
Conductor $6800$
Sign $1$
Analytic cond. $54.2982$
Root an. cond. $7.36873$
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  − 3-s − 2·9-s + 6·11-s + 3·13-s + 17-s + 7·19-s + 8·23-s + 5·27-s − 5·29-s − 5·31-s − 6·33-s + 8·37-s − 3·39-s + 4·43-s − 3·47-s − 7·49-s − 51-s + 9·53-s − 7·57-s − 5·59-s − 3·61-s + 2·67-s − 8·69-s + 15·71-s − 11·73-s − 8·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s + 1.80·11-s + 0.832·13-s + 0.242·17-s + 1.60·19-s + 1.66·23-s + 0.962·27-s − 0.928·29-s − 0.898·31-s − 1.04·33-s + 1.31·37-s − 0.480·39-s + 0.609·43-s − 0.437·47-s − 49-s − 0.140·51-s + 1.23·53-s − 0.927·57-s − 0.650·59-s − 0.384·61-s + 0.244·67-s − 0.963·69-s + 1.78·71-s − 1.28·73-s − 0.900·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6800\)    =    \(2^{4} \cdot 5^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(54.2982\)
Root analytic conductor: \(7.36873\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.030785949\)
\(L(\frac12)\) \(\approx\) \(2.030785949\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−7.926601570360659905393686576299, −7.11033404411954517136920129013, −6.56703116243100281481974387791, −5.78194421338694940250400840342, −5.35747152531346205554458017033, −4.36625093855695586860216711865, −3.55460953572539867577454990542, −2.93953932873213555178552008607, −1.49166526297850565867500440545, −0.842399540651489755749403345325, 0.842399540651489755749403345325, 1.49166526297850565867500440545, 2.93953932873213555178552008607, 3.55460953572539867577454990542, 4.36625093855695586860216711865, 5.35747152531346205554458017033, 5.78194421338694940250400840342, 6.56703116243100281481974387791, 7.11033404411954517136920129013, 7.926601570360659905393686576299

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