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 − 4·11-s + 2·13-s + 14-s + 16-s − 2·17-s − 20-s − 4·22-s + 8·23-s + 25-s + 2·26-s + 28-s + 6·29-s + 8·31-s + 32-s − 2·34-s − 35-s + 2·37-s − 40-s + 2·41-s − 12·43-s − 4·44-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 − 1.20·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s + 0.328·37-s − 0.158·40-s + 0.312·41-s − 1.82·43-s − 0.603·44-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: $-1$
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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.19178338210166, −12.86823068458375, −12.18881707049595, −11.80152457869026, −11.45688017251627, −10.75742922343316, −10.62198288162279, −10.14178412968821, −9.428445660382907, −8.806361699777899, −8.303535134005230, −8.097362179009306, −7.395361150115569, −6.897016399019519, −6.587645813786041, −5.844349007840462, −5.362780109554719, −4.916095846623809, −4.410680499177293, −4.048686025699837, −3.148232093855754, −2.827298968766981, −2.412571788105576, −1.421569940687072, −0.9495682615918058, 0, 0.9495682615918058, 1.421569940687072, 2.412571788105576, 2.827298968766981, 3.148232093855754, 4.048686025699837, 4.410680499177293, 4.916095846623809, 5.362780109554719, 5.844349007840462, 6.587645813786041, 6.897016399019519, 7.395361150115569, 8.097362179009306, 8.303535134005230, 8.806361699777899, 9.428445660382907, 10.14178412968821, 10.62198288162279, 10.75742922343316, 11.45688017251627, 11.80152457869026, 12.18881707049595, 12.86823068458375, 13.19178338210166

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