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Properties

Label 2-182070-1.1-c1-0-12
Degree $2$
Conductor $182070$
Sign $1$
Analytic cond. $1453.83$
Root an. cond. $38.1292$
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 − 2·13-s − 14-s + 16-s + 20-s − 4·22-s − 4·23-s + 25-s − 2·26-s − 28-s − 2·29-s + 32-s − 35-s − 2·37-s + 40-s − 6·41-s + 8·43-s − 4·44-s − 4·46-s − 8·47-s + 49-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 − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.223·20-s − 0.852·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s + 0.176·32-s − 0.169·35-s − 0.328·37-s + 0.158·40-s − 0.937·41-s + 1.21·43-s − 0.603·44-s − 0.589·46-s − 1.16·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1453.83\)
Root analytic conductor: \(38.1292\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 182070,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.755902806\)
\(L(\frac12)\) \(\approx\) \(1.755902806\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
17 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−13.17718368758228, −12.73531313856382, −12.36995600664942, −11.85602759136559, −11.28948165222574, −10.82342880918087, −10.29902344563048, −9.924817968115254, −9.552758550797973, −8.831289619769373, −8.307645031225770, −7.739646233854987, −7.360181086165983, −6.779724014882393, −6.250068677351284, −5.713600030048419, −5.380955490379285, −4.733824037776790, −4.389251181445980, −3.514322388150827, −3.169552533051191, −2.448951325388460, −2.099975714917971, −1.361441129226942, −0.3134880528016037, 0.3134880528016037, 1.361441129226942, 2.099975714917971, 2.448951325388460, 3.169552533051191, 3.514322388150827, 4.389251181445980, 4.733824037776790, 5.380955490379285, 5.713600030048419, 6.250068677351284, 6.779724014882393, 7.360181086165983, 7.739646233854987, 8.307645031225770, 8.831289619769373, 9.552758550797973, 9.924817968115254, 10.29902344563048, 10.82342880918087, 11.28948165222574, 11.85602759136559, 12.36995600664942, 12.73531313856382, 13.17718368758228

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