Properties

Degree $2$
Conductor $294$
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 + 8-s + 9-s + 2·10-s − 4·11-s + 12-s − 6·13-s + 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s + 2·20-s − 4·22-s + 8·23-s + 24-s − 25-s − 6·26-s + 27-s − 2·29-s + 2·30-s + 32-s − 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s − 1.66·13-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.852·22-s + 1.66·23-s + 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.371·29-s + 0.365·30-s + 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{294} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.384179895\)
\(L(\frac12)\) \(\approx\) \(2.384179895\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−19.49049772087496, −18.61945653539874, −17.63616747329080, −16.94000720632437, −15.81222876971925, −15.10375970384406, −14.31679686019816, −13.50387217119509, −12.94755124350849, −12.02200729864094, −10.75802487641203, −9.933813551342153, −9.086022258671444, −7.685321917971027, −6.954701945914613, −5.479902169847668, −4.828579559483360, −3.084812572455223, −2.160633482172023, 2.160633482172023, 3.084812572455223, 4.828579559483360, 5.479902169847668, 6.954701945914613, 7.685321917971027, 9.086022258671444, 9.933813551342153, 10.75802487641203, 12.02200729864094, 12.94755124350849, 13.50387217119509, 14.31679686019816, 15.10375970384406, 15.81222876971925, 16.94000720632437, 17.63616747329080, 18.61945653539874, 19.49049772087496

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