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Properties

Label 2-28322-1.1-c1-0-10
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 − 5-s − 3·6-s + 8-s + 6·9-s − 10-s − 3·12-s − 13-s + 3·15-s + 16-s + 6·18-s − 4·19-s − 20-s − 3·23-s − 3·24-s − 4·25-s − 26-s − 9·27-s − 6·29-s + 3·30-s − 6·31-s + 32-s + 6·36-s + 12·37-s − 4·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s + 0.353·8-s + 2·9-s − 0.316·10-s − 0.866·12-s − 0.277·13-s + 0.774·15-s + 1/4·16-s + 1.41·18-s − 0.917·19-s − 0.223·20-s − 0.625·23-s − 0.612·24-s − 4/5·25-s − 0.196·26-s − 1.73·27-s − 1.11·29-s + 0.547·30-s − 1.07·31-s + 0.176·32-s + 36-s + 1.97·37-s − 0.648·38-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 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−15.58307104648234, −14.89354218502667, −14.65388490860641, −13.68314816403027, −13.23171555482122, −12.59348002361850, −12.23445337865016, −11.88041981627377, −11.14065907625481, −10.82889457996704, −10.54119773376549, −9.510980109782635, −9.246345702418710, −8.012955483599851, −7.524253058699356, −7.133702365132730, −6.188134627178139, −5.957103919906467, −5.518287441268721, −4.659006600134827, −4.178558193273984, −3.828876056876325, −2.596610695180901, −1.873620081637274, −0.8460112572829445, 0, 0.8460112572829445, 1.873620081637274, 2.596610695180901, 3.828876056876325, 4.178558193273984, 4.659006600134827, 5.518287441268721, 5.957103919906467, 6.188134627178139, 7.133702365132730, 7.524253058699356, 8.012955483599851, 9.246345702418710, 9.510980109782635, 10.54119773376549, 10.82889457996704, 11.14065907625481, 11.88041981627377, 12.23445337865016, 12.59348002361850, 13.23171555482122, 13.68314816403027, 14.65388490860641, 14.89354218502667, 15.58307104648234

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