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Properties

Label 2-227430-1.1-c1-0-30
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 − 2·11-s − 6·13-s − 14-s + 16-s − 3·17-s − 20-s + 2·22-s + 2·23-s + 25-s + 6·26-s + 28-s − 4·31-s − 32-s + 3·34-s − 35-s − 5·37-s + 40-s + 10·41-s + 43-s − 2·44-s − 2·46-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 − 0.603·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.223·20-s + 0.426·22-s + 0.417·23-s + 1/5·25-s + 1.17·26-s + 0.188·28-s − 0.718·31-s − 0.176·32-s + 0.514·34-s − 0.169·35-s − 0.821·37-s + 0.158·40-s + 1.56·41-s + 0.152·43-s − 0.301·44-s − 0.294·46-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

\(L(1)\) \(\approx\) \(1.025628139\)
\(L(\frac12)\) \(\approx\) \(1.025628139\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−12.78986622688002, −12.50954365307298, −11.92700660231939, −11.41870997921087, −11.10400687118193, −10.61012273469481, −10.10297093873454, −9.702771023429388, −9.126916672705265, −8.792889669162934, −8.103926336067236, −7.845048615846299, −7.215602490372611, −7.022190001372291, −6.441775901102419, −5.581958301238643, −5.263933346972872, −4.719700064096802, −4.170480182668656, −3.526036869522976, −2.826618731438231, −2.245962175545773, −2.003023292398562, −0.9127779516598507, −0.3766546536535380, 0.3766546536535380, 0.9127779516598507, 2.003023292398562, 2.245962175545773, 2.826618731438231, 3.526036869522976, 4.170480182668656, 4.719700064096802, 5.263933346972872, 5.581958301238643, 6.441775901102419, 7.022190001372291, 7.215602490372611, 7.845048615846299, 8.103926336067236, 8.792889669162934, 9.126916672705265, 9.702771023429388, 10.10297093873454, 10.61012273469481, 11.10400687118193, 11.41870997921087, 11.92700660231939, 12.50954365307298, 12.78986622688002

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