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Properties

Label 2-227430-1.1-c1-0-138
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 4·13-s − 14-s + 16-s + 8·17-s − 20-s − 2·23-s + 25-s + 4·26-s + 28-s + 6·29-s + 6·31-s − 32-s − 8·34-s − 35-s + 2·37-s + 40-s + 6·41-s + 4·43-s + 2·46-s − 4·47-s + 49-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 − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.94·17-s − 0.223·20-s − 0.417·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s + 1.11·29-s + 1.07·31-s − 0.176·32-s − 1.37·34-s − 0.169·35-s + 0.328·37-s + 0.158·40-s + 0.937·41-s + 0.609·43-s + 0.294·46-s − 0.583·47-s + 1/7·49-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: \(1\)
Selberg data: \((2,\ 227430,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.91007532102838, −12.61944848492545, −12.05634790452060, −11.81383751925094, −11.41271580665745, −10.75349933204031, −10.21383089285414, −9.921288199699506, −9.624784805391198, −8.866235315474798, −8.375294418357844, −8.014469207822995, −7.542977538587946, −7.230850436579645, −6.615841692516874, −5.992947911705604, −5.508299349575264, −4.975176663564323, −4.402940390178058, −3.853092125495055, −3.141705345836314, −2.646533130317379, −2.172070495302346, −1.145707652642529, −0.9360605590973657, 0, 0.9360605590973657, 1.145707652642529, 2.172070495302346, 2.646533130317379, 3.141705345836314, 3.853092125495055, 4.402940390178058, 4.975176663564323, 5.508299349575264, 5.992947911705604, 6.615841692516874, 7.230850436579645, 7.542977538587946, 8.014469207822995, 8.375294418357844, 8.866235315474798, 9.624784805391198, 9.921288199699506, 10.21383089285414, 10.75349933204031, 11.41271580665745, 11.81383751925094, 12.05634790452060, 12.61944848492545, 12.91007532102838

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