Properties

Label 2-286110-1.1-c1-0-32
Degree $2$
Conductor $286110$
Sign $1$
Analytic cond. $2284.59$
Root an. cond. $47.7974$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 11-s + 2·14-s + 16-s − 20-s + 22-s − 4·23-s + 25-s + 2·28-s + 2·29-s − 4·31-s + 32-s − 2·35-s + 2·37-s − 40-s − 6·43-s + 44-s − 4·46-s − 3·49-s + 50-s + 6·53-s − 55-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.534·14-s + 1/4·16-s − 0.223·20-s + 0.213·22-s − 0.834·23-s + 1/5·25-s + 0.377·28-s + 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.338·35-s + 0.328·37-s − 0.158·40-s − 0.914·43-s + 0.150·44-s − 0.589·46-s − 3/7·49-s + 0.141·50-s + 0.824·53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(286110\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2284.59\)
Root analytic conductor: \(47.7974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{286110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 286110,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.176417497\)
\(L(\frac12)\) \(\approx\) \(3.176417497\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−12.82579252853734, −12.13325856403634, −11.84273485723813, −11.53187056263804, −11.00025172044418, −10.55419473774557, −10.13265860646285, −9.493890213813858, −9.018935180516036, −8.399853211056408, −8.039957970216789, −7.597664269261459, −7.095865762551403, −6.543566613415774, −6.127885093267568, −5.485231988509189, −5.079714621633498, −4.561602177768372, −4.027301823704362, −3.716102143893180, −2.963468824251327, −2.507761643140568, −1.674022584181179, −1.406730209779831, −0.4042557466429566, 0.4042557466429566, 1.406730209779831, 1.674022584181179, 2.507761643140568, 2.963468824251327, 3.716102143893180, 4.027301823704362, 4.561602177768372, 5.079714621633498, 5.485231988509189, 6.127885093267568, 6.543566613415774, 7.095865762551403, 7.597664269261459, 8.039957970216789, 8.399853211056408, 9.018935180516036, 9.493890213813858, 10.13265860646285, 10.55419473774557, 11.00025172044418, 11.53187056263804, 11.84273485723813, 12.13325856403634, 12.82579252853734

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