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Properties

Label 2-189630-1.1-c1-0-73
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 + 4·11-s − 4·13-s + 16-s + 4·17-s − 4·19-s − 20-s − 4·22-s − 8·23-s + 25-s + 4·26-s − 6·29-s + 4·31-s − 32-s − 4·34-s + 2·37-s + 4·38-s + 40-s + 10·41-s − 43-s + 4·44-s + 8·46-s + 4·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.10·13-s + 1/4·16-s + 0.970·17-s − 0.917·19-s − 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.784·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 1.56·41-s − 0.152·43-s + 0.603·44-s + 1.17·46-s + 0.583·47-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.35565728988100, −12.49820110098806, −12.28731046666096, −12.08832442466371, −11.42558777001278, −10.94230256885927, −10.58226747802614, −9.794701051392445, −9.606787967654182, −9.252555170603271, −8.522602024638255, −8.034728564750429, −7.714729249256818, −7.254189627722690, −6.569866229423823, −6.249071085263292, −5.680738775970682, −5.028472908029963, −4.333626836142437, −3.872583244604264, −3.471141500784110, −2.480598322183317, −2.214739109594210, −1.407019692648773, −0.7355263022448482, 0, 0.7355263022448482, 1.407019692648773, 2.214739109594210, 2.480598322183317, 3.471141500784110, 3.872583244604264, 4.333626836142437, 5.028472908029963, 5.680738775970682, 6.249071085263292, 6.569866229423823, 7.254189627722690, 7.714729249256818, 8.034728564750429, 8.522602024638255, 9.252555170603271, 9.606787967654182, 9.794701051392445, 10.58226747802614, 10.94230256885927, 11.42558777001278, 12.08832442466371, 12.28731046666096, 12.49820110098806, 13.35565728988100

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