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 + 2·17-s + 4·19-s − 21-s + 25-s + 27-s − 2·29-s + 8·31-s + 4·33-s − 35-s − 2·37-s − 2·39-s + 2·41-s + 4·43-s + 45-s + 49-s + 2·51-s − 10·53-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 + 0.485·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s − 0.169·35-s − 0.328·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.280·51-s − 1.37·53-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

\(L(1)\) \(\approx\) \(2.128934407\)
\(L(\frac12)\) \(\approx\) \(2.128934407\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.90314429446407, −19.15621008206790, −18.69305610141832, −17.57987209117578, −17.16636246713717, −16.27906897402989, −15.57428108308601, −14.65237796061494, −14.12677159418107, −13.52763422191635, −12.52231040716216, −11.96522030544318, −10.97251803449518, −9.773694967480984, −9.586914029366875, −8.621109740730131, −7.638261703317764, −6.786150261216271, −5.927922091040661, −4.771314870817133, −3.667982368759661, −2.688771284023954, −1.311387666113090, 1.311387666113090, 2.688771284023954, 3.667982368759661, 4.771314870817133, 5.927922091040661, 6.786150261216271, 7.638261703317764, 8.621109740730131, 9.586914029366875, 9.773694967480984, 10.97251803449518, 11.96522030544318, 12.52231040716216, 13.52763422191635, 14.12677159418107, 14.65237796061494, 15.57428108308601, 16.27906897402989, 17.16636246713717, 17.57987209117578, 18.69305610141832, 19.15621008206790, 19.90314429446407

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