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Properties

Label 2-182070-1.1-c1-0-65
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 2·11-s + 4·13-s + 14-s + 16-s + 6·19-s − 20-s − 2·22-s − 4·23-s + 25-s − 4·26-s − 28-s − 7·29-s − 3·31-s − 32-s + 35-s + 5·37-s − 6·38-s + 40-s + 2·41-s + 12·43-s + 2·44-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 + 0.603·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.37·19-s − 0.223·20-s − 0.426·22-s − 0.834·23-s + 1/5·25-s − 0.784·26-s − 0.188·28-s − 1.29·29-s − 0.538·31-s − 0.176·32-s + 0.169·35-s + 0.821·37-s − 0.973·38-s + 0.158·40-s + 0.312·41-s + 1.82·43-s + 0.301·44-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: \(1\)
Selberg data: \((2,\ 182070,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 15 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−13.13064014363808, −13.02875257397297, −12.34971765341383, −11.83766338133082, −11.39172173540808, −11.10250786062492, −10.62993388631490, −9.919060665144571, −9.512435052951274, −9.199570099315854, −8.679531744179758, −8.106543705209325, −7.576524752768271, −7.368236012436134, −6.609202285737294, −6.173427664555585, −5.710558671291259, −5.200049869563905, −4.219272659740105, −3.926380050390681, −3.339647862391833, −2.830426684305460, −2.009521212456179, −1.361597690447918, −0.8279120819531727, 0, 0.8279120819531727, 1.361597690447918, 2.009521212456179, 2.830426684305460, 3.339647862391833, 3.926380050390681, 4.219272659740105, 5.200049869563905, 5.710558671291259, 6.173427664555585, 6.609202285737294, 7.368236012436134, 7.576524752768271, 8.106543705209325, 8.679531744179758, 9.199570099315854, 9.512435052951274, 9.919060665144571, 10.62993388631490, 11.10250786062492, 11.39172173540808, 11.83766338133082, 12.34971765341383, 13.02875257397297, 13.13064014363808

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