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 − 2·13-s + 14-s + 16-s + 6·17-s + 20-s + 25-s − 2·26-s + 28-s − 6·29-s + 4·31-s + 32-s + 6·34-s + 35-s − 2·37-s + 40-s + 6·41-s + 8·43-s + 12·47-s + 49-s + 50-s − 2·52-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 − 1.11·29-s + 0.718·31-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 + 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.141·50-s − 0.277·52-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

\(L(1)\) \(\approx\) \(6.327317116\)
\(L(\frac12)\) \(\approx\) \(6.327317116\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.72121128545601, −12.59920866910647, −12.08974164924511, −11.72754624382430, −11.00257464094006, −10.77711358670904, −10.13033673868454, −9.829815991512093, −9.109966981314272, −8.905618615969073, −7.964188671627284, −7.642446475170360, −7.351627951552827, −6.657933695680804, −6.020594577986113, −5.708317160052387, −5.240690238220330, −4.743824714840001, −4.113641214212279, −3.680619602014272, −2.984865858159950, −2.477490294039163, −1.960028505662451, −1.212556129005527, −0.6431913033408858, 0.6431913033408858, 1.212556129005527, 1.960028505662451, 2.477490294039163, 2.984865858159950, 3.680619602014272, 4.113641214212279, 4.743824714840001, 5.240690238220330, 5.708317160052387, 6.020594577986113, 6.657933695680804, 7.351627951552827, 7.642446475170360, 7.964188671627284, 8.905618615969073, 9.109966981314272, 9.829815991512093, 10.13033673868454, 10.77711358670904, 11.00257464094006, 11.72754624382430, 12.08974164924511, 12.59920866910647, 12.72121128545601

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