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Properties

Label 2-227430-1.1-c1-0-95
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 + 6·11-s + 6·13-s + 14-s + 16-s + 3·17-s − 20-s − 6·22-s − 6·23-s + 25-s − 6·26-s − 28-s − 4·29-s + 4·31-s − 32-s − 3·34-s + 35-s + 7·37-s + 40-s − 2·41-s + 11·43-s + 6·44-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 + 1.80·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.223·20-s − 1.27·22-s − 1.25·23-s + 1/5·25-s − 1.17·26-s − 0.188·28-s − 0.742·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s + 0.169·35-s + 1.15·37-s + 0.158·40-s − 0.312·41-s + 1.67·43-s + 0.904·44-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: $1$
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

\(L(1)\) \(\approx\) \(2.891643731\)
\(L(\frac12)\) \(\approx\) \(2.891643731\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−12.84105913951776, −12.32980852148875, −11.86192492459010, −11.52749094156914, −11.20091084569711, −10.45022615095005, −10.26609787865618, −9.477131110029765, −9.183785200183103, −8.825253308154904, −8.250821621953242, −7.815825520151584, −7.333157876233010, −6.708951861221226, −6.271245971893933, −5.934041974688427, −5.481818418346290, −4.327780682121132, −4.057522607868163, −3.641235616151762, −3.106214700239712, −2.272382773196274, −1.655427632282096, −0.8975541713784926, −0.7188428165465057, 0.7188428165465057, 0.8975541713784926, 1.655427632282096, 2.272382773196274, 3.106214700239712, 3.641235616151762, 4.057522607868163, 4.327780682121132, 5.481818418346290, 5.934041974688427, 6.271245971893933, 6.708951861221226, 7.333157876233010, 7.815825520151584, 8.250821621953242, 8.825253308154904, 9.183785200183103, 9.477131110029765, 10.26609787865618, 10.45022615095005, 11.20091084569711, 11.52749094156914, 11.86192492459010, 12.32980852148875, 12.84105913951776

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