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Properties

Label 2-106470-1.1-c1-0-66
Degree $2$
Conductor $106470$
Sign $-1$
Analytic cond. $850.167$
Root an. cond. $29.1576$
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 − 7-s − 8-s + 10-s + 14-s + 16-s − 2·19-s − 20-s + 25-s − 28-s − 6·29-s − 8·31-s − 32-s + 35-s + 4·37-s + 2·38-s + 40-s + 6·41-s + 2·43-s − 6·47-s + 49-s − 50-s − 6·53-s + 56-s + 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.267·14-s + 1/4·16-s − 0.458·19-s − 0.223·20-s + 1/5·25-s − 0.188·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.169·35-s + 0.657·37-s + 0.324·38-s + 0.158·40-s + 0.937·41-s + 0.304·43-s − 0.875·47-s + 1/7·49-s − 0.141·50-s − 0.824·53-s + 0.133·56-s + 0.787·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(106470\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(850.167\)
Root analytic conductor: \(29.1576\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 106470,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.11273468006250, −13.15721648020299, −12.85053633812952, −12.68124185922858, −11.69578594981935, −11.51527299462941, −11.01148798011945, −10.47506558527555, −10.00445224951617, −9.366366648407334, −9.110528143195786, −8.505755733660001, −7.949914450372735, −7.505074935219355, −7.034523170037611, −6.498185924992880, −5.884684797759875, −5.422285981953433, −4.692144734608098, −3.935200759338762, −3.607375193027190, −2.830722246364591, −2.215403989369252, −1.572544843857877, −0.6968882446250804, 0, 0.6968882446250804, 1.572544843857877, 2.215403989369252, 2.830722246364591, 3.607375193027190, 3.935200759338762, 4.692144734608098, 5.422285981953433, 5.884684797759875, 6.498185924992880, 7.034523170037611, 7.505074935219355, 7.949914450372735, 8.505755733660001, 9.110528143195786, 9.366366648407334, 10.00445224951617, 10.47506558527555, 11.01148798011945, 11.51527299462941, 11.69578594981935, 12.68124185922858, 12.85053633812952, 13.15721648020299, 14.11273468006250

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