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Properties

Label 2-227430-1.1-c1-0-71
Degree $2$
Conductor $227430$
Sign $-1$
Analytic cond. $1816.03$
Root an. cond. $42.6149$
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 − 6·13-s − 14-s + 16-s + 5·17-s − 20-s + 2·22-s + 3·23-s + 25-s + 6·26-s + 28-s − 4·29-s − 2·31-s − 32-s − 5·34-s − 35-s + 37-s + 40-s − 11·41-s − 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 − 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.223·20-s + 0.426·22-s + 0.625·23-s + 1/5·25-s + 1.17·26-s + 0.188·28-s − 0.742·29-s − 0.359·31-s − 0.176·32-s − 0.857·34-s − 0.169·35-s + 0.164·37-s + 0.158·40-s − 1.71·41-s − 0.152·43-s − 0.301·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(227430\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(1816.03\)
Root analytic conductor: \(42.6149\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 227430,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
19 \( 1 \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.08706995574050, −12.53295746590785, −12.16236381256932, −11.77361467820914, −11.34119602803154, −10.72837922899028, −10.35872394832665, −9.886258537539894, −9.507649577565351, −8.979323416161964, −8.335894014133506, −8.029906894191775, −7.460051817561749, −7.210160902601311, −6.786979772565597, −5.889630564386870, −5.467772384037882, −4.968103042872307, −4.551080143762600, −3.737333256685380, −3.138803281134517, −2.735283786483790, −2.000450447256038, −1.491622102918128, −0.6273377716625716, 0, 0.6273377716625716, 1.491622102918128, 2.000450447256038, 2.735283786483790, 3.138803281134517, 3.737333256685380, 4.551080143762600, 4.968103042872307, 5.467772384037882, 5.889630564386870, 6.786979772565597, 7.210160902601311, 7.460051817561749, 8.029906894191775, 8.335894014133506, 8.979323416161964, 9.507649577565351, 9.886258537539894, 10.35872394832665, 10.72837922899028, 11.34119602803154, 11.77361467820914, 12.16236381256932, 12.53295746590785, 13.08706995574050

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