Properties

Label 2-221067-1.1-c1-0-1
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-s − 4-s − 2·5-s + 7-s − 3·8-s − 2·10-s + 2·13-s + 14-s − 16-s − 2·17-s + 2·20-s + 8·23-s − 25-s + 2·26-s − 28-s − 29-s + 8·31-s + 5·32-s − 2·34-s − 2·35-s − 10·37-s + 6·40-s + 6·41-s − 4·43-s + 8·46-s − 8·47-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s − 1.06·8-s − 0.632·10-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.447·20-s + 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.185·29-s + 1.43·31-s + 0.883·32-s − 0.342·34-s − 0.338·35-s − 1.64·37-s + 0.948·40-s + 0.937·41-s − 0.609·43-s + 1.17·46-s − 1.16·47-s + 1/7·49-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

\(L(1)\) \(\approx\) \(1.118320892\)
\(L(\frac12)\) \(\approx\) \(1.118320892\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−12.95941411223670, −12.55007602091958, −12.08486633580825, −11.68465659633247, −11.10673112010813, −10.92898816343916, −10.22785129340286, −9.546958253952118, −9.221763848691819, −8.643624465645973, −8.175750575576071, −7.979678151961419, −7.135578790687587, −6.707428815425471, −6.258425282364558, −5.524393769778360, −5.135656439445232, −4.557094455703096, −4.278967435153747, −3.681827327355622, −3.054326744103598, −2.849294776378736, −1.737056827696230, −1.153929802537112, −0.2834423582652990, 0.2834423582652990, 1.153929802537112, 1.737056827696230, 2.849294776378736, 3.054326744103598, 3.681827327355622, 4.278967435153747, 4.557094455703096, 5.135656439445232, 5.524393769778360, 6.258425282364558, 6.707428815425471, 7.135578790687587, 7.979678151961419, 8.175750575576071, 8.643624465645973, 9.221763848691819, 9.546958253952118, 10.22785129340286, 10.92898816343916, 11.10673112010813, 11.68465659633247, 12.08486633580825, 12.55007602091958, 12.95941411223670

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