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Properties

Label 2-106470-1.1-c1-0-46
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 4·11-s + 14-s + 16-s − 2·17-s − 20-s + 4·22-s + 25-s + 28-s − 6·29-s − 8·31-s + 32-s − 2·34-s − 35-s + 10·37-s − 40-s + 2·41-s + 4·43-s + 4·44-s + 8·47-s + 49-s + 50-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 + 1.20·11-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s + 1.64·37-s − 0.158·40-s + 0.312·41-s + 0.609·43-s + 0.603·44-s + 1.16·47-s + 1/7·49-s + 0.141·50-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: \(0\)
Selberg data: \((2,\ 106470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.015682861\)
\(L(\frac12)\) \(\approx\) \(4.015682861\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.69969896330064, −13.14656393348143, −12.78480265957558, −12.19887962681614, −11.75278626548104, −11.39162903274628, −10.79607119277263, −10.63038817538029, −9.610752837601570, −9.192993819360401, −8.891074425153005, −8.094879952530152, −7.448913882318934, −7.331041400852917, −6.551759203007708, −5.991561548130466, −5.659992233145725, −4.848941215097962, −4.299485746339784, −3.995626625492932, −3.393131297192485, −2.696959619161915, −1.966397007711453, −1.409617585743309, −0.5535736427399809, 0.5535736427399809, 1.409617585743309, 1.966397007711453, 2.696959619161915, 3.393131297192485, 3.995626625492932, 4.299485746339784, 4.848941215097962, 5.659992233145725, 5.991561548130466, 6.551759203007708, 7.331041400852917, 7.448913882318934, 8.094879952530152, 8.891074425153005, 9.192993819360401, 9.610752837601570, 10.63038817538029, 10.79607119277263, 11.39162903274628, 11.75278626548104, 12.19887962681614, 12.78480265957558, 13.14656393348143, 13.69969896330064

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