Properties

Label 2-222180-1.1-c1-0-7
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s + 9-s + 11-s + 4·13-s + 15-s − 6·17-s − 7·19-s + 21-s + 25-s − 27-s − 6·29-s + 4·31-s − 33-s + 35-s + 2·37-s − 4·39-s + 9·41-s − 2·43-s − 45-s + 7·47-s + 49-s + 6·51-s − 5·53-s − 55-s + 7·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.258·15-s − 1.45·17-s − 1.60·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s + 0.169·35-s + 0.328·37-s − 0.640·39-s + 1.40·41-s − 0.304·43-s − 0.149·45-s + 1.02·47-s + 1/7·49-s + 0.840·51-s − 0.686·53-s − 0.134·55-s + 0.927·57-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: $\chi_{222180} (1, \cdot )$
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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.17927137166538, −12.64057080414671, −12.45146668316865, −11.58443088044780, −11.32638558063551, −10.93892996896942, −10.62166544934950, −9.949943413262520, −9.465358669181464, −8.840790625117116, −8.565869580930900, −8.130159153165242, −7.287187071391950, −7.015531729383833, −6.394285127978084, −6.053236480964000, −5.698646789614962, −4.763569900245837, −4.406739521899256, −3.915925246267528, −3.554585221400674, −2.582953717180320, −2.186524943288300, −1.395966451927177, −0.6615844048850001, 0, 0.6615844048850001, 1.395966451927177, 2.186524943288300, 2.582953717180320, 3.554585221400674, 3.915925246267528, 4.406739521899256, 4.763569900245837, 5.698646789614962, 6.053236480964000, 6.394285127978084, 7.015531729383833, 7.287187071391950, 8.130159153165242, 8.565869580930900, 8.840790625117116, 9.465358669181464, 9.949943413262520, 10.62166544934950, 10.93892996896942, 11.32638558063551, 11.58443088044780, 12.45146668316865, 12.64057080414671, 13.17927137166538

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