Properties

Label 2-8400-1.1-c1-0-13
Degree $2$
Conductor $8400$
Sign $1$
Analytic cond. $67.0743$
Root an. cond. $8.18989$
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 − 7-s + 9-s − 2·11-s − 6·13-s − 4·17-s + 6·19-s − 21-s − 8·23-s + 27-s + 6·29-s + 2·31-s − 2·33-s − 4·37-s − 6·39-s + 2·41-s + 4·43-s + 8·47-s + 49-s − 4·51-s − 6·53-s + 6·57-s + 8·59-s − 10·61-s − 63-s + 8·67-s − 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 0.970·17-s + 1.37·19-s − 0.218·21-s − 1.66·23-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.348·33-s − 0.657·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.560·51-s − 0.824·53-s + 0.794·57-s + 1.04·59-s − 1.28·61-s − 0.125·63-s + 0.977·67-s − 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(67.0743\)
Root analytic conductor: \(8.18989\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.724464814\)
\(L(\frac12)\) \(\approx\) \(1.724464814\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
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

−7.80635544827580026450461952153, −7.20721281106445443151687816362, −6.56827409134700813810895743988, −5.66839572149385890964068143306, −4.92793348079080743273636716161, −4.29757174685115541700192382531, −3.37307891705985744977489192290, −2.56887503190621219638583502612, −2.07618167487879001187013162426, −0.59274365801986098627370066764, 0.59274365801986098627370066764, 2.07618167487879001187013162426, 2.56887503190621219638583502612, 3.37307891705985744977489192290, 4.29757174685115541700192382531, 4.92793348079080743273636716161, 5.66839572149385890964068143306, 6.56827409134700813810895743988, 7.20721281106445443151687816362, 7.80635544827580026450461952153

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