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Properties

Label 2-28322-1.1-c1-0-20
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 − 3·3-s + 4-s + 2·5-s + 3·6-s − 8-s + 6·9-s − 2·10-s + 5·11-s − 3·12-s − 3·13-s − 6·15-s + 16-s − 6·18-s − 2·19-s + 2·20-s − 5·22-s + 8·23-s + 3·24-s − 25-s + 3·26-s − 9·27-s + 6·29-s + 6·30-s + 4·31-s − 32-s − 15·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.894·5-s + 1.22·6-s − 0.353·8-s + 2·9-s − 0.632·10-s + 1.50·11-s − 0.866·12-s − 0.832·13-s − 1.54·15-s + 1/4·16-s − 1.41·18-s − 0.458·19-s + 0.447·20-s − 1.06·22-s + 1.66·23-s + 0.612·24-s − 1/5·25-s + 0.588·26-s − 1.73·27-s + 1.11·29-s + 1.09·30-s + 0.718·31-s − 0.176·32-s − 2.61·33-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 \( 1 + p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−15.54667053238164, −15.16178592007377, −14.44896368315954, −13.90226319738163, −13.16305295462636, −12.61487808358425, −12.04289059034639, −11.67462156800226, −11.26722420056244, −10.44098466427033, −10.20930914482944, −9.712238049665095, −8.991950968134067, −8.633432499339089, −7.527667794249223, −6.939114419938888, −6.515468420696310, −6.216976068697017, −5.390980598059635, −4.941668437430210, −4.312663633717197, −3.336844835954814, −2.340944161535095, −1.470009709747453, −1.022115358516072, 0, 1.022115358516072, 1.470009709747453, 2.340944161535095, 3.336844835954814, 4.312663633717197, 4.941668437430210, 5.390980598059635, 6.216976068697017, 6.515468420696310, 6.939114419938888, 7.527667794249223, 8.633432499339089, 8.991950968134067, 9.712238049665095, 10.20930914482944, 10.44098466427033, 11.26722420056244, 11.67462156800226, 12.04289059034639, 12.61487808358425, 13.16305295462636, 13.90226319738163, 14.44896368315954, 15.16178592007377, 15.54667053238164

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