Properties

Label 2-286650-1.1-c1-0-150
Degree $2$
Conductor $286650$
Sign $-1$
Analytic cond. $2288.91$
Root an. cond. $47.8425$
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 − 8-s − 6·11-s + 13-s + 16-s − 7·17-s + 7·19-s + 6·22-s − 8·23-s − 26-s + 2·29-s − 8·31-s − 32-s + 7·34-s + 8·37-s − 7·38-s + 3·41-s + 7·43-s − 6·44-s + 8·46-s + 10·47-s + 52-s − 12·53-s − 2·58-s − 4·59-s − 12·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 0.277·13-s + 1/4·16-s − 1.69·17-s + 1.60·19-s + 1.27·22-s − 1.66·23-s − 0.196·26-s + 0.371·29-s − 1.43·31-s − 0.176·32-s + 1.20·34-s + 1.31·37-s − 1.13·38-s + 0.468·41-s + 1.06·43-s − 0.904·44-s + 1.17·46-s + 1.45·47-s + 0.138·52-s − 1.64·53-s − 0.262·58-s − 0.520·59-s − 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(286650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(2288.91\)
Root analytic conductor: \(47.8425\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 286650,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−12.96258893521440, −12.43400563719467, −12.04108637197756, −11.41220052218581, −10.96253755302025, −10.72560997680824, −10.25815011344569, −9.625413224099656, −9.339287875063226, −8.843769476436528, −8.272034267045430, −7.812180942713141, −7.381089770776356, −7.247094768440624, −6.185897137563571, −5.994933985145816, −5.537526323364988, −4.800350016924547, −4.392143027171391, −3.760391768883945, −2.898607652470832, −2.708606268206647, −2.025511274338347, −1.488250136202198, −0.5583791755810921, 0, 0.5583791755810921, 1.488250136202198, 2.025511274338347, 2.708606268206647, 2.898607652470832, 3.760391768883945, 4.392143027171391, 4.800350016924547, 5.537526323364988, 5.994933985145816, 6.185897137563571, 7.247094768440624, 7.381089770776356, 7.812180942713141, 8.272034267045430, 8.843769476436528, 9.339287875063226, 9.625413224099656, 10.25815011344569, 10.72560997680824, 10.96253755302025, 11.41220052218581, 12.04108637197756, 12.43400563719467, 12.96258893521440

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