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Properties

Label 2-28322-1.1-c1-0-19
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-s + 4-s − 4·5-s + 6-s + 8-s − 2·9-s − 4·10-s + 12-s + 2·13-s − 4·15-s + 16-s − 2·18-s − 3·19-s − 4·20-s + 4·23-s + 24-s + 11·25-s + 2·26-s − 5·27-s + 29-s − 4·30-s + 9·31-s + 32-s − 2·36-s − 10·37-s − 3·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 1.26·10-s + 0.288·12-s + 0.554·13-s − 1.03·15-s + 1/4·16-s − 0.471·18-s − 0.688·19-s − 0.894·20-s + 0.834·23-s + 0.204·24-s + 11/5·25-s + 0.392·26-s − 0.962·27-s + 0.185·29-s − 0.730·30-s + 1.61·31-s + 0.176·32-s − 1/3·36-s − 1.64·37-s − 0.486·38-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 \( 1 - T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−15.40658995317888, −15.00002437725638, −14.47195421879319, −13.95924506908532, −13.36461096619447, −12.86907462723579, −12.12044496609660, −11.84664754590541, −11.38347566366009, −10.70648494880758, −10.41816337400147, −9.279188066723282, −8.751822110473534, −8.151784777523541, −8.008375812472990, −7.122860924136481, −6.682550268370274, −6.013181055113079, −5.061584779564589, −4.659356013281225, −3.900874371298057, −3.403680289241160, −3.001191145318931, −2.148707120889617, −1.012195564024959, 0, 1.012195564024959, 2.148707120889617, 3.001191145318931, 3.403680289241160, 3.900874371298057, 4.659356013281225, 5.061584779564589, 6.013181055113079, 6.682550268370274, 7.122860924136481, 8.008375812472990, 8.151784777523541, 8.751822110473534, 9.279188066723282, 10.41816337400147, 10.70648494880758, 11.38347566366009, 11.84664754590541, 12.12044496609660, 12.86907462723579, 13.36461096619447, 13.95924506908532, 14.47195421879319, 15.00002437725638, 15.40658995317888

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