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Properties

Label 2-227430-1.1-c1-0-61
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 $0$

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 + 20-s + 25-s + 2·26-s + 28-s + 6·29-s − 8·31-s − 32-s + 35-s + 4·37-s − 40-s + 6·41-s + 2·43-s + 6·47-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 + 0.223·20-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 + 0.169·35-s + 0.657·37-s − 0.158·40-s + 0.937·41-s + 0.304·43-s + 0.875·47-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: $1$
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: \(0\)
Selberg data: \((2,\ 227430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.536397485\)
\(L(\frac12)\) \(\approx\) \(2.536397485\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−12.84550444654968, −12.38705360301421, −12.04346809455494, −11.40471220380213, −11.02014925855190, −10.58470923336739, −10.11451024957851, −9.599122218841993, −9.295393123322741, −8.682368847442899, −8.334815132036827, −7.757128094041067, −7.210045245157608, −6.978317656333258, −6.189298539149724, −5.865692909842928, −5.181375604460696, −4.855484495300289, −4.021682206175812, −3.626483720659748, −2.704291314633907, −2.355106067590249, −1.856687957729025, −0.9794926709738582, −0.5842393217332040, 0.5842393217332040, 0.9794926709738582, 1.856687957729025, 2.355106067590249, 2.704291314633907, 3.626483720659748, 4.021682206175812, 4.855484495300289, 5.181375604460696, 5.865692909842928, 6.189298539149724, 6.978317656333258, 7.210045245157608, 7.757128094041067, 8.334815132036827, 8.682368847442899, 9.295393123322741, 9.599122218841993, 10.11451024957851, 10.58470923336739, 11.02014925855190, 11.40471220380213, 12.04346809455494, 12.38705360301421, 12.84550444654968

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