Properties

Degree $2$
Conductor $714$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2·5-s − 6-s + 7-s + 8-s + 9-s − 2·10-s + 4·11-s − 12-s − 2·13-s + 14-s + 2·15-s + 16-s + 17-s + 18-s + 4·19-s − 2·20-s − 21-s + 4·22-s + 8·23-s − 24-s − 25-s − 2·26-s − 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{714} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.882709516\)
\(L(\frac12)\) \(\approx\) \(1.882709516\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.49709984358206, −19.09918264461832, −17.93319951702142, −17.24912203383099, −16.53099618796036, −15.85002131625542, −15.04073351424746, −14.48711258335164, −13.63982832617044, −12.62595052388836, −11.89325394972863, −11.56932410934060, −10.73554355414389, −9.670342250781385, −8.640309548688445, −7.470290332942099, −6.957673813778495, −5.842089710148834, −4.846280376868273, −4.113070354319421, −3.005488617646356, −1.179421882253824, 1.179421882253824, 3.005488617646356, 4.113070354319421, 4.846280376868273, 5.842089710148834, 6.957673813778495, 7.470290332942099, 8.640309548688445, 9.670342250781385, 10.73554355414389, 11.56932410934060, 11.89325394972863, 12.62595052388836, 13.63982832617044, 14.48711258335164, 15.04073351424746, 15.85002131625542, 16.53099618796036, 17.24912203383099, 17.93319951702142, 19.09918264461832, 19.49709984358206

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