Properties

Label 2-28322-1.1-c1-0-30
Degree $2$
Conductor $28322$
Sign $-1$
Analytic cond. $226.152$
Root an. cond. $15.0383$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2·5-s + 8-s − 3·9-s + 2·10-s + 2·13-s + 16-s − 3·18-s − 4·19-s + 2·20-s − 25-s + 2·26-s + 6·29-s + 32-s − 3·36-s + 6·37-s − 4·38-s + 2·40-s − 6·41-s − 12·43-s − 6·45-s − 8·47-s − 50-s + 2·52-s − 2·53-s + 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s − 9-s + 0.632·10-s + 0.554·13-s + 1/4·16-s − 0.707·18-s − 0.917·19-s + 0.447·20-s − 1/5·25-s + 0.392·26-s + 1.11·29-s + 0.176·32-s − 1/2·36-s + 0.986·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s − 1.82·43-s − 0.894·45-s − 1.16·47-s − 0.141·50-s + 0.277·52-s − 0.274·53-s + 0.787·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28322\)    =    \(2 \cdot 7^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(226.152\)
Root analytic conductor: \(15.0383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 28322,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.34220483437642, −14.80876264289998, −14.37005070231958, −13.81317129100496, −13.43310946579989, −12.99986322270227, −12.30291167402746, −11.79226512460799, −11.16329603366486, −10.83952667827270, −9.972501520803618, −9.744258686034757, −8.839346559880173, −8.293807603166725, −7.967445317669717, −6.685763271351715, −6.605673582410295, −5.926695384849067, −5.371752605210680, −4.834561939190213, −4.059448972057715, −3.311977933432902, −2.708090491492663, −2.030471001545682, −1.294387649284134, 0, 1.294387649284134, 2.030471001545682, 2.708090491492663, 3.311977933432902, 4.059448972057715, 4.834561939190213, 5.371752605210680, 5.926695384849067, 6.605673582410295, 6.685763271351715, 7.967445317669717, 8.293807603166725, 8.839346559880173, 9.744258686034757, 9.972501520803618, 10.83952667827270, 11.16329603366486, 11.79226512460799, 12.30291167402746, 12.99986322270227, 13.43310946579989, 13.81317129100496, 14.37005070231958, 14.80876264289998, 15.34220483437642

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