Properties

Degree $2$
Conductor $840$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 7-s + 9-s + 4·11-s − 2·13-s − 15-s − 6·17-s + 4·19-s − 21-s + 8·23-s + 25-s − 27-s − 2·29-s − 4·33-s + 35-s − 2·37-s + 2·39-s + 10·41-s + 4·43-s + 45-s + 49-s + 6·51-s + 14·53-s + 4·55-s − 4·57-s + 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.169·35-s − 0.328·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.840·51-s + 1.92·53-s + 0.539·55-s − 0.529·57-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{840} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.527981036\)
\(L(\frac12)\) \(\approx\) \(1.527981036\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−19.75268090486595, −19.28251371617830, −18.22208501878643, −17.68235986623116, −17.11702116067083, −16.51980448285787, −15.56542954548869, −14.82546266466826, −14.13255502615022, −13.27917175823007, −12.57573024644847, −11.58043669136490, −11.20676224859790, −10.24217028435075, −9.260421204459755, −8.814898165377068, −7.341354642903596, −6.811207620989472, −5.789420309892787, −4.912682255938004, −3.996971914267832, −2.480230925178014, −1.113200828171494, 1.113200828171494, 2.480230925178014, 3.996971914267832, 4.912682255938004, 5.789420309892787, 6.811207620989472, 7.341354642903596, 8.814898165377068, 9.260421204459755, 10.24217028435075, 11.20676224859790, 11.58043669136490, 12.57573024644847, 13.27917175823007, 14.13255502615022, 14.82546266466826, 15.56542954548869, 16.51980448285787, 17.11702116067083, 17.68235986623116, 18.22208501878643, 19.28251371617830, 19.75268090486595

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