Properties

Label 2-166410-1.1-c1-0-22
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 + 4·7-s + 8-s − 10-s + 4·11-s + 4·13-s + 4·14-s + 16-s − 4·17-s − 4·19-s − 20-s + 4·22-s − 8·23-s + 25-s + 4·26-s + 4·28-s + 6·29-s − 4·31-s + 32-s − 4·34-s − 4·35-s − 2·37-s − 4·38-s − 40-s − 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 1.10·13-s + 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.917·19-s − 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.784·26-s + 0.755·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.685·34-s − 0.676·35-s − 0.328·37-s − 0.648·38-s − 0.158·40-s − 1.56·41-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 166410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.249583633\)
\(L(\frac12)\) \(\approx\) \(5.249583633\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.40665317551619, −12.72912955731660, −12.12358460691484, −11.89137852626337, −11.33195389344490, −11.12395324321480, −10.59598006300286, −10.07241460700438, −9.375669014421465, −8.574045715148531, −8.383306325828700, −8.197820754998350, −7.277826422190017, −6.816139696548886, −6.396593402968976, −5.879803775593373, −5.254412375095637, −4.613101082520201, −4.303286755920876, −3.788775675131747, −3.415981373897205, −2.335267731020194, −1.874912667338656, −1.449177948895119, −0.5780992828847850, 0.5780992828847850, 1.449177948895119, 1.874912667338656, 2.335267731020194, 3.415981373897205, 3.788775675131747, 4.303286755920876, 4.613101082520201, 5.254412375095637, 5.879803775593373, 6.396593402968976, 6.816139696548886, 7.277826422190017, 8.197820754998350, 8.383306325828700, 8.574045715148531, 9.375669014421465, 10.07241460700438, 10.59598006300286, 11.12395324321480, 11.33195389344490, 11.89137852626337, 12.12358460691484, 12.72912955731660, 13.40665317551619

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