Properties

Label 2-166410-1.1-c1-0-27
Degree $2$
Conductor $166410$
Sign $1$
Analytic cond. $1328.79$
Root an. cond. $36.4525$
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 + 2·7-s + 8-s + 10-s + 3·11-s − 4·13-s + 2·14-s + 16-s + 3·17-s − 19-s + 20-s + 3·22-s + 6·23-s + 25-s − 4·26-s + 2·28-s − 6·29-s − 4·31-s + 32-s + 3·34-s + 2·35-s + 8·37-s − 38-s + 40-s + 9·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.904·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.223·20-s + 0.639·22-s + 1.25·23-s + 1/5·25-s − 0.784·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s + 0.338·35-s + 1.31·37-s − 0.162·38-s + 0.158·40-s + 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(166410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(1328.79\)
Root analytic conductor: \(36.4525\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{166410} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 166410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.300268965\)
\(L(\frac12)\) \(\approx\) \(6.300268965\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
43 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.25602377157253, −12.80697248918639, −12.25324416677563, −11.96280779853486, −11.38242515317869, −10.91022495376148, −10.63323007507663, −9.819583168536234, −9.413178542040097, −9.098485171570649, −8.383917405904034, −7.768019397369977, −7.258290331918873, −7.056397726138306, −6.256730069531848, −5.712802871455056, −5.410835112709584, −4.784578135393826, −4.259487270917390, −3.883524364925361, −2.972363782298788, −2.643438037168341, −1.863568145388710, −1.386684289480718, −0.6456907424571472, 0.6456907424571472, 1.386684289480718, 1.863568145388710, 2.643438037168341, 2.972363782298788, 3.883524364925361, 4.259487270917390, 4.784578135393826, 5.410835112709584, 5.712802871455056, 6.256730069531848, 7.056397726138306, 7.258290331918873, 7.768019397369977, 8.383917405904034, 9.098485171570649, 9.413178542040097, 9.819583168536234, 10.63323007507663, 10.91022495376148, 11.38242515317869, 11.96280779853486, 12.25324416677563, 12.80697248918639, 13.25602377157253

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