Properties

Label 2-221067-1.1-c1-0-11
Degree $2$
Conductor $221067$
Sign $1$
Analytic cond. $1765.22$
Root an. cond. $42.0146$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 4·5-s + 7-s − 8·10-s − 2·13-s − 2·14-s − 4·16-s − 5·17-s + 5·19-s + 8·20-s + 11·25-s + 4·26-s + 2·28-s − 29-s − 3·31-s + 8·32-s + 10·34-s + 4·35-s − 8·37-s − 10·38-s + 10·41-s − 4·43-s + 8·47-s + 49-s − 22·50-s − 4·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.78·5-s + 0.377·7-s − 2.52·10-s − 0.554·13-s − 0.534·14-s − 16-s − 1.21·17-s + 1.14·19-s + 1.78·20-s + 11/5·25-s + 0.784·26-s + 0.377·28-s − 0.185·29-s − 0.538·31-s + 1.41·32-s + 1.71·34-s + 0.676·35-s − 1.31·37-s − 1.62·38-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 3.11·50-s − 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221067\)    =    \(3^{2} \cdot 7 \cdot 11^{2} \cdot 29\)
Sign: $1$
Analytic conductor: \(1765.22\)
Root analytic conductor: \(42.0146\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{221067} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 221067,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.656125361\)
\(L(\frac12)\) \(\approx\) \(1.656125361\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
29 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.13241786493711, −12.42847873611449, −11.97908843145085, −11.27095791636905, −10.91582378974465, −10.48092221265804, −10.04962107988087, −9.628039854754199, −9.246800713172216, −8.829739428053616, −8.569865172937410, −7.755300903181731, −7.219185403450700, −7.033688159752555, −6.344658681449318, −5.761266114556169, −5.412428962391457, −4.729384763875793, −4.358137350635683, −3.344151896901376, −2.614300007179398, −2.189625851090071, −1.716737582048357, −1.201707566356128, −0.4638896077784511, 0.4638896077784511, 1.201707566356128, 1.716737582048357, 2.189625851090071, 2.614300007179398, 3.344151896901376, 4.358137350635683, 4.729384763875793, 5.412428962391457, 5.761266114556169, 6.344658681449318, 7.033688159752555, 7.219185403450700, 7.755300903181731, 8.569865172937410, 8.829739428053616, 9.246800713172216, 9.628039854754199, 10.04962107988087, 10.48092221265804, 10.91582378974465, 11.27095791636905, 11.97908843145085, 12.42847873611449, 13.13241786493711

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