Properties

Label 2-189630-1.1-c1-0-127
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 8-s + 10-s + 2·11-s + 2·13-s + 16-s − 4·17-s + 6·19-s + 20-s + 2·22-s − 6·23-s + 25-s + 2·26-s + 10·29-s + 8·31-s + 32-s − 4·34-s + 2·37-s + 6·38-s + 40-s + 2·41-s − 43-s + 2·44-s − 6·46-s − 2·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 0.603·11-s + 0.554·13-s + 1/4·16-s − 0.970·17-s + 1.37·19-s + 0.223·20-s + 0.426·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 1.85·29-s + 1.43·31-s + 0.176·32-s − 0.685·34-s + 0.328·37-s + 0.973·38-s + 0.158·40-s + 0.312·41-s − 0.152·43-s + 0.301·44-s − 0.884·46-s − 0.291·47-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: \(1\)
Selberg data: \((2,\ 189630,\ (\ :1/2),\ -1)\)

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 \( 1 \)
43 \( 1 + T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.35501424115304, −13.08184848425551, −12.33495135637256, −11.94919300104427, −11.52864723343728, −11.27796741517843, −10.34194244600348, −10.20459214422116, −9.704820337551586, −9.032999901917924, −8.584791672099778, −8.121138613290175, −7.499736424571289, −6.947273012680609, −6.385606906499327, −6.132778341222181, −5.632623248743064, −4.911631929766750, −4.428443864157054, −4.112739821736905, −3.287094133688472, −2.825245330201031, −2.345478131284805, −1.332328009906875, −1.236156409410615, 0, 1.236156409410615, 1.332328009906875, 2.345478131284805, 2.825245330201031, 3.287094133688472, 4.112739821736905, 4.428443864157054, 4.911631929766750, 5.632623248743064, 6.132778341222181, 6.385606906499327, 6.947273012680609, 7.499736424571289, 8.121138613290175, 8.584791672099778, 9.032999901917924, 9.704820337551586, 10.20459214422116, 10.34194244600348, 11.27796741517843, 11.52864723343728, 11.94919300104427, 12.33495135637256, 13.08184848425551, 13.35501424115304

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