Properties

Label 2-166410-1.1-c1-0-42
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 $1$

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: \(1\)
Selberg data: \((2,\ 166410,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
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

−13.57908111102009, −12.71655008299261, −12.41950751843647, −12.06463983033401, −11.70402998509796, −10.85219723205771, −10.74102454109446, −10.05549858177775, −9.561908628108435, −9.186582323768466, −8.891202878550162, −8.107251988997298, −7.726431856516933, −7.120092937329299, −6.809369481143580, −6.398529402737917, −5.576161834702736, −5.191243381353917, −4.495946088603070, −3.817399071913439, −3.341877898517549, −2.768972275740528, −2.228526654055936, −1.278438019146812, −0.8070180577359259, 0, 0.8070180577359259, 1.278438019146812, 2.228526654055936, 2.768972275740528, 3.341877898517549, 3.817399071913439, 4.495946088603070, 5.191243381353917, 5.576161834702736, 6.398529402737917, 6.809369481143580, 7.120092937329299, 7.726431856516933, 8.107251988997298, 8.891202878550162, 9.186582323768466, 9.561908628108435, 10.05549858177775, 10.74102454109446, 10.85219723205771, 11.70402998509796, 12.06463983033401, 12.41950751843647, 12.71655008299261, 13.57908111102009

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