↑

Properties

Label 2-222180-1.1-c1-0-19
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 + 5·11-s + 4·13-s − 15-s − 2·17-s − 5·19-s + 21-s + 25-s + 27-s − 2·29-s + 8·31-s + 5·33-s − 35-s + 6·37-s + 4·39-s − 41-s − 10·43-s − 45-s − 7·47-s + 49-s − 2·51-s − 53-s − 5·55-s − 5·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.10·13-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 − 0.371·29-s + 1.43·31-s + 0.870·33-s − 0.169·35-s + 0.986·37-s + 0.640·39-s − 0.156·41-s − 1.52·43-s − 0.149·45-s − 1.02·47-s + 1/7·49-s − 0.280·51-s − 0.137·53-s − 0.674·55-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 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−13.30121184469367, −12.81507838529266, −12.17712720033787, −11.77986308884317, −11.32241206655129, −11.02658389574499, −10.36520348482367, −9.941748294850277, −9.235998316873252, −8.974040667980841, −8.407007621890226, −8.189844888749393, −7.637143714057625, −6.861831300029270, −6.555256483928882, −6.191177378370343, −5.544352046740306, −4.598361620512009, −4.323551648819154, −4.044886356750207, −3.159171028754147, −3.003986943401952, −1.903127671763899, −1.597357489194850, −0.9615573247706417, 0, 0.9615573247706417, 1.597357489194850, 1.903127671763899, 3.003986943401952, 3.159171028754147, 4.044886356750207, 4.323551648819154, 4.598361620512009, 5.544352046740306, 6.191177378370343, 6.555256483928882, 6.861831300029270, 7.637143714057625, 8.189844888749393, 8.407007621890226, 8.974040667980841, 9.235998316873252, 9.941748294850277, 10.36520348482367, 11.02658389574499, 11.32241206655129, 11.77986308884317, 12.17712720033787, 12.81507838529266, 13.30121184469367

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