Properties

Label 2-223440-1.1-c1-0-187
Degree $2$
Conductor $223440$
Sign $1$
Analytic cond. $1784.17$
Root an. cond. $42.2395$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s − 4·11-s − 2·13-s − 15-s − 2·17-s − 19-s − 4·23-s + 25-s − 27-s + 6·29-s + 4·31-s + 4·33-s − 6·37-s + 2·39-s − 10·41-s + 4·43-s + 45-s − 12·47-s + 2·51-s + 6·53-s − 4·55-s + 57-s − 12·59-s + 2·61-s − 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.696·33-s − 0.986·37-s + 0.320·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s + 0.280·51-s + 0.824·53-s − 0.539·55-s + 0.132·57-s − 1.56·59-s + 0.256·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 223440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(223440\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1784.17\)
Root analytic conductor: \(42.2395\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{223440} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 223440,\ (\ :1/2),\ 1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
19 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.36275227988941, −13.03086759418915, −12.33658301830912, −12.13011892221618, −11.64285249179573, −11.00535106409227, −10.54260290933576, −10.22986515354095, −9.849202345169801, −9.344984430186001, −8.585191638154197, −8.271604553263394, −7.792314984503147, −7.140675259656202, −6.661267287849286, −6.324791262254361, −5.593259695151190, −5.310048061828834, −4.666639950562919, −4.428449169051126, −3.551969945185265, −2.887973773054819, −2.455123750638075, −1.785560737559392, −1.196955540771391, 0, 0, 1.196955540771391, 1.785560737559392, 2.455123750638075, 2.887973773054819, 3.551969945185265, 4.428449169051126, 4.666639950562919, 5.310048061828834, 5.593259695151190, 6.324791262254361, 6.661267287849286, 7.140675259656202, 7.792314984503147, 8.271604553263394, 8.585191638154197, 9.344984430186001, 9.849202345169801, 10.22986515354095, 10.54260290933576, 11.00535106409227, 11.64285249179573, 12.13011892221618, 12.33658301830912, 13.03086759418915, 13.36275227988941

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