Properties

Degree $2$
Conductor $336$
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 − 2·5-s + 7-s + 9-s + 4·11-s + 6·13-s − 2·15-s + 2·17-s + 4·19-s + 21-s − 8·23-s − 25-s + 27-s − 2·29-s + 4·33-s − 2·35-s − 10·37-s + 6·39-s − 6·41-s + 4·43-s − 2·45-s + 49-s + 2·51-s + 6·53-s − 8·55-s + 4·57-s − 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.516·15-s + 0.485·17-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 − 0.338·35-s − 1.64·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 1.07·55-s + 0.529·57-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{336} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.576986771\)
\(L(\frac12)\) \(\approx\) \(1.576986771\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−19.87827133896883, −19.06847623697571, −18.36110848031421, −17.53237502214832, −16.33973891717598, −15.83106190801461, −15.00414631018725, −14.03891478248381, −13.60989589251896, −12.14741682098256, −11.72901697799890, −10.71603142151067, −9.567727627901465, −8.579197535720293, −7.963562405403294, −6.894997659600310, −5.697617718889750, −4.047524811413720, −3.540529749465783, −1.508006199521713, 1.508006199521713, 3.540529749465783, 4.047524811413720, 5.697617718889750, 6.894997659600310, 7.963562405403294, 8.579197535720293, 9.567727627901465, 10.71603142151067, 11.72901697799890, 12.14741682098256, 13.60989589251896, 14.03891478248381, 15.00414631018725, 15.83106190801461, 16.33973891717598, 17.53237502214832, 18.36110848031421, 19.06847623697571, 19.87827133896883

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