↑

Properties

Label 2-222180-1.1-c1-0-18
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 − 2·11-s + 4·13-s − 15-s − 2·17-s + 2·19-s + 21-s + 25-s + 27-s − 2·29-s − 6·31-s − 2·33-s − 35-s + 6·37-s + 4·39-s + 6·41-s + 4·43-s − 45-s + 49-s − 2·51-s − 8·53-s + 2·55-s + 2·57-s + 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.258·15-s − 0.485·17-s + 0.458·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.07·31-s − 0.348·33-s − 0.169·35-s + 0.986·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s − 1.09·53-s + 0.269·55-s + 0.264·57-s + 1.28·61-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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

−13.24737860811705, −12.74596364822641, −12.47514573217174, −11.57456130649149, −11.37823020892888, −10.88989445055005, −10.53296848074163, −9.864254387504772, −9.309010968648171, −8.993462249015000, −8.432517499024529, −7.972226716979215, −7.643038504574942, −7.150207778491695, −6.543333738445992, −5.970336391039690, −5.474534813375169, −4.888514383386816, −4.325424843925444, −3.772675779199217, −3.449562249099389, −2.633617125546584, −2.270164479842244, −1.449676260194961, −0.9120078012972046, 0, 0.9120078012972046, 1.449676260194961, 2.270164479842244, 2.633617125546584, 3.449562249099389, 3.772675779199217, 4.325424843925444, 4.888514383386816, 5.474534813375169, 5.970336391039690, 6.543333738445992, 7.150207778491695, 7.643038504574942, 7.972226716979215, 8.432517499024529, 8.993462249015000, 9.309010968648171, 9.864254387504772, 10.53296848074163, 10.88989445055005, 11.37823020892888, 11.57456130649149, 12.47514573217174, 12.74596364822641, 13.24737860811705

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