Properties

Degree $2$
Conductor $227430$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 5·11-s + 14-s + 16-s − 2·17-s − 20-s − 5·22-s + 2·23-s + 25-s − 28-s − 10·29-s + 2·31-s − 32-s + 2·34-s + 35-s − 4·37-s + 40-s + 5·41-s + 4·43-s + 5·44-s − 2·46-s − 4·47-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 + 1.50·11-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s − 1.06·22-s + 0.417·23-s + 1/5·25-s − 0.188·28-s − 1.85·29-s + 0.359·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.657·37-s + 0.158·40-s + 0.780·41-s + 0.609·43-s + 0.753·44-s − 0.294·46-s − 0.583·47-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: \(0\)
Selberg data: \((2,\ 227430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6402039187\)
\(L(\frac12)\) \(\approx\) \(0.6402039187\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−12.73035437295922, −12.42817765257322, −11.98598873457014, −11.31481303864058, −11.19342183615773, −10.72580030627670, −10.01374575953871, −9.533621616773966, −9.177360861750109, −8.885836070229752, −8.267716090055854, −7.735359733357666, −7.284710807785129, −6.693136660437872, −6.528821787482697, −5.798268969220084, −5.355993955266248, −4.503880869427381, −4.051013546422049, −3.605557079417278, −2.988800294756903, −2.374692843368804, −1.549918833393030, −1.234174189538865, −0.2588192866674218, 0.2588192866674218, 1.234174189538865, 1.549918833393030, 2.374692843368804, 2.988800294756903, 3.605557079417278, 4.051013546422049, 4.503880869427381, 5.355993955266248, 5.798268969220084, 6.528821787482697, 6.693136660437872, 7.284710807785129, 7.735359733357666, 8.267716090055854, 8.885836070229752, 9.177360861750109, 9.533621616773966, 10.01374575953871, 10.72580030627670, 11.19342183615773, 11.31481303864058, 11.98598873457014, 12.42817765257322, 12.73035437295922

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