Properties

Label 2-162-1.1-c1-0-0
Degree $2$
Conductor $162$
Sign $1$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
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 + 2·7-s − 8-s + 3·11-s + 2·13-s − 2·14-s + 16-s + 3·17-s − 19-s − 3·22-s + 6·23-s − 5·25-s − 2·26-s + 2·28-s − 6·29-s − 4·31-s − 32-s − 3·34-s − 4·37-s + 38-s − 9·41-s − 43-s + 3·44-s − 6·46-s + 6·47-s − 3·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.904·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s − 0.639·22-s + 1.25·23-s − 25-s − 0.392·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s − 0.657·37-s + 0.162·38-s − 1.40·41-s − 0.152·43-s + 0.452·44-s − 0.884·46-s + 0.875·47-s − 3/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9335592475\)
\(L(\frac12)\) \(\approx\) \(0.9335592475\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−12.70434390882727936880586234897, −11.57619177441088474085986142761, −10.96157598355786499811507828582, −9.711604924732818951217103858923, −8.789487803962222669402730529275, −7.80670854475898716623947697687, −6.68619713403218599725876916726, −5.34383976963704470462884839275, −3.65076060485342096132488694617, −1.60693251106342657124988453680, 1.60693251106342657124988453680, 3.65076060485342096132488694617, 5.34383976963704470462884839275, 6.68619713403218599725876916726, 7.80670854475898716623947697687, 8.789487803962222669402730529275, 9.711604924732818951217103858923, 10.96157598355786499811507828582, 11.57619177441088474085986142761, 12.70434390882727936880586234897

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