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Properties

Label 2-106470-1.1-c1-0-128
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

−14.03809875478216, −13.35862535178467, −12.94537691514713, −12.69061818348674, −11.99771035785758, −11.65059453475390, −11.02877317487891, −10.45352952985851, −10.20048543622519, −9.558565775146528, −8.936053520660269, −8.603639151581526, −7.858556601782477, −7.237343225021334, −6.911886830478762, −6.143559247617319, −5.936965878602199, −5.278874204145305, −4.706893510995757, −4.161785909109949, −3.572797590890997, −2.935989511896404, −2.412408721342645, −1.750402748067800, −1.014819220842457, 0, 1.014819220842457, 1.750402748067800, 2.412408721342645, 2.935989511896404, 3.572797590890997, 4.161785909109949, 4.706893510995757, 5.278874204145305, 5.936965878602199, 6.143559247617319, 6.911886830478762, 7.237343225021334, 7.858556601782477, 8.603639151581526, 8.936053520660269, 9.558565775146528, 10.20048543622519, 10.45352952985851, 11.02877317487891, 11.65059453475390, 11.99771035785758, 12.69061818348674, 12.94537691514713, 13.35862535178467, 14.03809875478216

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