Properties

Label 2-227430-1.1-c1-0-26
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

\(L(1)\) \(\approx\) \(2.052203875\)
\(L(\frac12)\) \(\approx\) \(2.052203875\)
\(L(\frac{3}{2})\) 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

−12.96543014183252, −12.54271524477858, −12.02428488001543, −11.56241079822403, −11.22677459302050, −10.74180751953583, −10.32215582627263, −9.590379941484310, −9.287915983335001, −8.675834341583134, −8.075126694045329, −7.541550646653955, −7.415051344526514, −6.639175000142097, −6.267213150314588, −5.548780503396564, −5.184660265937405, −4.682144326954850, −4.170807924484809, −3.562001126930553, −3.216154094099329, −2.437229636914841, −1.894899040670876, −1.338754601024280, −0.3359590102481302, 0.3359590102481302, 1.338754601024280, 1.894899040670876, 2.437229636914841, 3.216154094099329, 3.562001126930553, 4.170807924484809, 4.682144326954850, 5.184660265937405, 5.548780503396564, 6.267213150314588, 6.639175000142097, 7.415051344526514, 7.541550646653955, 8.075126694045329, 8.675834341583134, 9.287915983335001, 9.590379941484310, 10.32215582627263, 10.74180751953583, 11.22677459302050, 11.56241079822403, 12.02428488001543, 12.54271524477858, 12.96543014183252

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