↑

Properties

Label 2-222180-1.1-c1-0-15
Degree $2$
Conductor $222180$
Sign $-1$
Analytic cond. $1774.11$
Root an. cond. $42.1202$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 7-s + 9-s + 11-s + 13-s − 15-s − 2·17-s − 5·19-s + 21-s + 25-s + 27-s + 8·29-s − 31-s + 33-s − 35-s − 2·37-s + 39-s − 2·41-s − 43-s − 45-s − 12·47-s + 49-s − 2·51-s − 2·53-s − 55-s − 5·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.258·15-s − 0.485·17-s − 1.14·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 0.179·31-s + 0.174·33-s − 0.169·35-s − 0.328·37-s + 0.160·39-s − 0.312·41-s − 0.152·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s − 0.134·55-s − 0.662·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(222180\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(1774.11\)
Root analytic conductor: \(42.1202\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 222180,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
23 \( 1 \)
good11 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 17 T + p T^{2} \)
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

−13.24675158096333, −12.72918657710181, −12.31087163261423, −11.78630030291502, −11.36905429347550, −10.79874917676533, −10.51487588586192, −9.866557929370984, −9.441414222510650, −8.789878956265090, −8.456241021011093, −8.138844145798714, −7.655000552505341, −6.865619217945835, −6.640085775019137, −6.186129134861402, −5.323708007079577, −4.852580932704205, −4.341694336043831, −3.906701701745613, −3.312316811318822, −2.763214864421396, −2.093767659745771, −1.597139240996863, −0.8428071016137437, 0, 0.8428071016137437, 1.597139240996863, 2.093767659745771, 2.763214864421396, 3.312316811318822, 3.906701701745613, 4.341694336043831, 4.852580932704205, 5.323708007079577, 6.186129134861402, 6.640085775019137, 6.865619217945835, 7.655000552505341, 8.138844145798714, 8.456241021011093, 8.789878956265090, 9.441414222510650, 9.866557929370984, 10.51487588586192, 10.79874917676533, 11.36905429347550, 11.78630030291502, 12.31087163261423, 12.72918657710181, 13.24675158096333

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