Properties

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

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 + 2·17-s − 4·19-s + 21-s + 25-s − 27-s − 10·29-s + 4·33-s − 35-s + 6·37-s + 2·39-s − 6·41-s − 4·43-s + 45-s − 8·47-s + 49-s − 2·51-s + 6·53-s − 4·55-s + 4·57-s − 4·59-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 + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.696·33-s − 0.169·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s − 0.520·59-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: \(1\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
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

−19.41907654628509, −18.60440292299219, −18.24238638590715, −17.27502979999820, −16.72790809882322, −16.13163216397358, −15.12838428782636, −14.68042464799634, −13.34095513794510, −13.10468601860796, −12.24351621882781, −11.35528156436744, −10.48462532291354, −9.975450018667805, −9.082259719636065, −7.955913714504645, −7.183293596317799, −6.141587047271160, −5.440602148971476, −4.534353275481180, −3.159142580960679, −1.939136604605553, 0, 1.939136604605553, 3.159142580960679, 4.534353275481180, 5.440602148971476, 6.141587047271160, 7.183293596317799, 7.955913714504645, 9.082259719636065, 9.975450018667805, 10.48462532291354, 11.35528156436744, 12.24351621882781, 13.10468601860796, 13.34095513794510, 14.68042464799634, 15.12838428782636, 16.13163216397358, 16.72790809882322, 17.27502979999820, 18.24238638590715, 18.60440292299219, 19.41907654628509

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