Properties

Label 2-221067-1.1-c1-0-13
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  − 2·4-s − 7-s − 2·13-s + 4·16-s + 6·17-s + 7·19-s + 3·23-s − 5·25-s + 2·28-s + 29-s − 4·31-s + 2·37-s + 9·41-s + 4·43-s + 9·47-s + 49-s + 4·52-s − 9·53-s + 6·59-s − 2·61-s − 8·64-s − 13·67-s − 12·68-s + 9·71-s + 73-s − 14·76-s + 10·79-s + ⋯
L(s)  = 1  − 4-s − 0.377·7-s − 0.554·13-s + 16-s + 1.45·17-s + 1.60·19-s + 0.625·23-s − 25-s + 0.377·28-s + 0.185·29-s − 0.718·31-s + 0.328·37-s + 1.40·41-s + 0.609·43-s + 1.31·47-s + 1/7·49-s + 0.554·52-s − 1.23·53-s + 0.781·59-s − 0.256·61-s − 64-s − 1.58·67-s − 1.45·68-s + 1.06·71-s + 0.117·73-s − 1.60·76-s + 1.12·79-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

\(L(1)\) \(\approx\) \(2.073306921\)
\(L(\frac12)\) \(\approx\) \(2.073306921\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−13.04929848556323, −12.35288369806719, −12.21990797993814, −11.77604240983777, −10.97931104650407, −10.63985124248019, −9.966435071753479, −9.566795413246591, −9.373706619205142, −8.943363851282010, −8.091211518813537, −7.725355340510044, −7.495946005815605, −6.845640133201566, −6.039296236981566, −5.590096928039181, −5.332207838418576, −4.732124112973169, −4.046012347383273, −3.689677935301811, −3.030852835080917, −2.662500301427963, −1.641640576054179, −0.9847643994885511, −0.5005198937189511, 0.5005198937189511, 0.9847643994885511, 1.641640576054179, 2.662500301427963, 3.030852835080917, 3.689677935301811, 4.046012347383273, 4.732124112973169, 5.332207838418576, 5.590096928039181, 6.039296236981566, 6.845640133201566, 7.495946005815605, 7.725355340510044, 8.091211518813537, 8.943363851282010, 9.373706619205142, 9.566795413246591, 9.966435071753479, 10.63985124248019, 10.97931104650407, 11.77604240983777, 12.21990797993814, 12.35288369806719, 13.04929848556323

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