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Properties

Label 2-189630-1.1-c1-0-71
Degree $2$
Conductor $189630$
Sign $1$
Analytic cond. $1514.20$
Root an. cond. $38.9127$
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 − 8-s + 10-s + 5·11-s + 4·13-s + 16-s − 2·19-s − 20-s − 5·22-s − 3·23-s + 25-s − 4·26-s + 6·29-s + 6·31-s − 32-s + 2·37-s + 2·38-s + 40-s + 9·41-s + 43-s + 5·44-s + 3·46-s + 8·47-s − 50-s + 4·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.50·11-s + 1.10·13-s + 1/4·16-s − 0.458·19-s − 0.223·20-s − 1.06·22-s − 0.625·23-s + 1/5·25-s − 0.784·26-s + 1.11·29-s + 1.07·31-s − 0.176·32-s + 0.328·37-s + 0.324·38-s + 0.158·40-s + 1.40·41-s + 0.152·43-s + 0.753·44-s + 0.442·46-s + 1.16·47-s − 0.141·50-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(1514.20\)
Root analytic conductor: \(38.9127\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 189630,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.769456437\)
\(L(\frac12)\) \(\approx\) \(2.769456437\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43 \( 1 - T \)
good11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−12.95212864514692, −12.55804609854984, −12.04351560428409, −11.58039832358119, −11.31849495219076, −10.74167734878919, −10.29725693759663, −9.740498159443817, −9.289064870207773, −8.746753144090030, −8.429143997733862, −8.016343867177791, −7.388127374072439, −6.795225659778563, −6.413787491513837, −6.055791260122387, −5.438021157544408, −4.537074883571229, −4.076571891863000, −3.775091422738693, −2.997690178251664, −2.396581720809925, −1.694865367471014, −0.9196810348960551, −0.7075679125940603, 0.7075679125940603, 0.9196810348960551, 1.694865367471014, 2.396581720809925, 2.997690178251664, 3.775091422738693, 4.076571891863000, 4.537074883571229, 5.438021157544408, 6.055791260122387, 6.413787491513837, 6.795225659778563, 7.388127374072439, 8.016343867177791, 8.429143997733862, 8.746753144090030, 9.289064870207773, 9.740498159443817, 10.29725693759663, 10.74167734878919, 11.31849495219076, 11.58039832358119, 12.04351560428409, 12.55804609854984, 12.95212864514692

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