Properties

Label 2-286110-1.1-c1-0-144
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s − 11-s + 6·13-s + 16-s + 4·19-s + 20-s + 22-s + 25-s − 6·26-s + 6·29-s − 32-s + 2·37-s − 4·38-s − 40-s − 6·41-s − 4·43-s − 44-s + 8·47-s − 7·49-s − 50-s + 6·52-s − 14·53-s − 55-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.66·13-s + 1/4·16-s + 0.917·19-s + 0.223·20-s + 0.213·22-s + 1/5·25-s − 1.17·26-s + 1.11·29-s − 0.176·32-s + 0.328·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.150·44-s + 1.16·47-s − 49-s − 0.141·50-s + 0.832·52-s − 1.92·53-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 286110,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−12.98802473032274, −12.37290178810672, −12.07394375522816, −11.35894335465788, −11.08323071991725, −10.68834554983276, −10.12894576961018, −9.771484935264986, −9.218721312995885, −8.877025363529398, −8.278580133688963, −7.980249790188521, −7.515300735771638, −6.732124425662105, −6.422036398712253, −6.085447785306785, −5.362412317947688, −5.018443094945242, −4.302803169856568, −3.601129622850711, −3.153077038091697, −2.681695423676577, −1.861061320697125, −1.360743843533922, −0.9078702215464194, 0, 0.9078702215464194, 1.360743843533922, 1.861061320697125, 2.681695423676577, 3.153077038091697, 3.601129622850711, 4.302803169856568, 5.018443094945242, 5.362412317947688, 6.085447785306785, 6.422036398712253, 6.732124425662105, 7.515300735771638, 7.980249790188521, 8.278580133688963, 8.877025363529398, 9.218721312995885, 9.771484935264986, 10.12894576961018, 10.68834554983276, 11.08323071991725, 11.35894335465788, 12.07394375522816, 12.37290178810672, 12.98802473032274

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