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Properties

Label 2-227430-1.1-c1-0-16
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 − 3·11-s − 6·13-s + 14-s + 16-s − 2·17-s − 20-s + 3·22-s + 23-s + 25-s + 6·26-s − 28-s + 29-s − 2·31-s − 32-s + 2·34-s + 35-s − 2·37-s + 40-s − 3·41-s + 7·43-s − 3·44-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 − 0.904·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s + 0.639·22-s + 0.208·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.185·29-s − 0.359·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.328·37-s + 0.158·40-s − 0.468·41-s + 1.06·43-s − 0.452·44-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

\(L(1)\) \(\approx\) \(0.5895167230\)
\(L(\frac12)\) \(\approx\) \(0.5895167230\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.91808190244486, −12.44186266675528, −11.93926449703259, −11.56020041730295, −11.01855459050227, −10.48852846971166, −10.14018198599974, −9.732083299011760, −9.155901711281767, −8.804699601286448, −8.134214282713762, −7.760329767354832, −7.340937865933046, −6.855595551330128, −6.478238200536209, −5.680510581445057, −5.191669369206050, −4.777176472095401, −4.131833922310945, −3.428225422678837, −2.909769949512471, −2.300046684685977, −1.988807736980323, −0.8817557617966769, −0.2901307390476283, 0.2901307390476283, 0.8817557617966769, 1.988807736980323, 2.300046684685977, 2.909769949512471, 3.428225422678837, 4.131833922310945, 4.777176472095401, 5.191669369206050, 5.680510581445057, 6.478238200536209, 6.855595551330128, 7.340937865933046, 7.760329767354832, 8.134214282713762, 8.804699601286448, 9.155901711281767, 9.732083299011760, 10.14018198599974, 10.48852846971166, 11.01855459050227, 11.56020041730295, 11.93926449703259, 12.44186266675528, 12.91808190244486

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