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Properties

Label 2-227430-1.1-c1-0-90
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 + 11-s − 2·13-s + 14-s + 16-s + 6·17-s − 20-s − 22-s − 5·23-s + 25-s + 2·26-s − 28-s − 9·29-s − 2·31-s − 32-s − 6·34-s + 35-s + 6·37-s + 40-s + 9·41-s − 3·43-s + 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 + 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.223·20-s − 0.213·22-s − 1.04·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s − 1.67·29-s − 0.359·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s + 0.986·37-s + 0.158·40-s + 1.40·41-s − 0.457·43-s + 0.150·44-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: \(1\)
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 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
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

−13.06552289196122, −12.50314982308456, −12.26401723852519, −11.73628540470765, −11.29871566224941, −10.84270987894235, −10.30260448006259, −9.799034519083836, −9.397282839878857, −9.204825484875903, −8.344087280878702, −7.835302028110652, −7.740093501723000, −7.106443280238532, −6.597148990837633, −6.023709103894163, −5.524679547244890, −5.123159432801341, −4.135951962490181, −3.891748374125278, −3.275611772461879, −2.652520717241933, −2.070377716250792, −1.376918529920710, −0.6936897746707956, 0, 0.6936897746707956, 1.376918529920710, 2.070377716250792, 2.652520717241933, 3.275611772461879, 3.891748374125278, 4.135951962490181, 5.123159432801341, 5.524679547244890, 6.023709103894163, 6.597148990837633, 7.106443280238532, 7.740093501723000, 7.835302028110652, 8.344087280878702, 9.204825484875903, 9.397282839878857, 9.799034519083836, 10.30260448006259, 10.84270987894235, 11.29871566224941, 11.73628540470765, 12.26401723852519, 12.50314982308456, 13.06552289196122

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