Properties

Degree $2$
Conductor $227430$
Sign $-1$
Motivic weight $1$
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 − 2·13-s − 14-s + 16-s − 6·17-s − 20-s + 25-s + 2·26-s + 28-s − 8·31-s − 32-s + 6·34-s − 35-s − 2·37-s + 40-s − 6·41-s − 10·43-s + 49-s − 50-s − 2·52-s + 6·53-s − 56-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 − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.169·35-s − 0.328·37-s + 0.158·40-s − 0.937·41-s − 1.52·43-s + 1/7·49-s − 0.141·50-s − 0.277·52-s + 0.824·53-s − 0.133·56-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{227430} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−13.15643418920940, −12.77856523263471, −11.91499272373831, −11.74092449052834, −11.41206066473032, −10.72323430779251, −10.44787911182579, −9.944850945879462, −9.344376105691518, −8.883068787283736, −8.482401176982721, −8.155678397222716, −7.408382847061734, −7.038124154652540, −6.804014231845094, −6.041047212198634, −5.454805078406646, −4.953464071324939, −4.409682288250665, −3.832019630503310, −3.260280073704520, −2.584991736690368, −1.962077998986172, −1.592013844966698, −0.5868301578925793, 0, 0.5868301578925793, 1.592013844966698, 1.962077998986172, 2.584991736690368, 3.260280073704520, 3.832019630503310, 4.409682288250665, 4.953464071324939, 5.454805078406646, 6.041047212198634, 6.804014231845094, 7.038124154652540, 7.408382847061734, 8.155678397222716, 8.482401176982721, 8.883068787283736, 9.344376105691518, 9.944850945879462, 10.44787911182579, 10.72323430779251, 11.41206066473032, 11.74092449052834, 11.91499272373831, 12.77856523263471, 13.15643418920940

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