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Properties

Label 2-221760-1.1-c1-0-271
Degree $2$
Conductor $221760$
Sign $-1$
Analytic cond. $1770.76$
Root an. cond. $42.0804$
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  + 5-s + 7-s − 11-s + 2·13-s − 2·17-s − 2·23-s + 25-s + 2·29-s + 2·31-s + 35-s − 2·37-s − 12·47-s + 49-s − 2·53-s − 55-s − 4·59-s + 2·61-s + 2·65-s − 12·67-s + 4·71-s + 6·73-s − 77-s + 10·79-s − 6·83-s − 2·85-s − 2·89-s + 2·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.417·23-s + 1/5·25-s + 0.371·29-s + 0.359·31-s + 0.169·35-s − 0.328·37-s − 1.75·47-s + 1/7·49-s − 0.274·53-s − 0.134·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s + 0.474·71-s + 0.702·73-s − 0.113·77-s + 1.12·79-s − 0.658·83-s − 0.216·85-s − 0.211·89-s + 0.209·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221760\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(1770.76\)
Root analytic conductor: \(42.0804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 221760,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.32247223646251, −12.73504397892992, −12.30373851208365, −11.75486753433652, −11.35819960542697, −10.70324588337214, −10.60446850499393, −9.860137723061623, −9.515064682463795, −8.978629506995114, −8.383262260599956, −8.131860706007828, −7.569623727269676, −6.925444171180167, −6.399036450783189, −6.147014582166293, −5.302405565112395, −5.127027742216414, −4.361027643016672, −4.000996923154886, −3.155229561448771, −2.816441010165667, −1.957653437833117, −1.638465982314484, −0.8413363234448568, 0, 0.8413363234448568, 1.638465982314484, 1.957653437833117, 2.816441010165667, 3.155229561448771, 4.000996923154886, 4.361027643016672, 5.127027742216414, 5.302405565112395, 6.147014582166293, 6.399036450783189, 6.925444171180167, 7.569623727269676, 8.131860706007828, 8.383262260599956, 8.978629506995114, 9.515064682463795, 9.860137723061623, 10.60446850499393, 10.70324588337214, 11.35819960542697, 11.75486753433652, 12.30373851208365, 12.73504397892992, 13.32247223646251

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