Properties

Label 2-221067-1.1-c1-0-10
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 − 3·5-s − 7-s − 3·8-s − 3·10-s + 3·13-s − 14-s − 16-s + 17-s + 6·19-s + 3·20-s − 6·23-s + 4·25-s + 3·26-s + 28-s + 29-s + 5·31-s + 5·32-s + 34-s + 3·35-s − 12·37-s + 6·38-s + 9·40-s + 2·41-s + 43-s − 6·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.34·5-s − 0.377·7-s − 1.06·8-s − 0.948·10-s + 0.832·13-s − 0.267·14-s − 1/4·16-s + 0.242·17-s + 1.37·19-s + 0.670·20-s − 1.25·23-s + 4/5·25-s + 0.588·26-s + 0.188·28-s + 0.185·29-s + 0.898·31-s + 0.883·32-s + 0.171·34-s + 0.507·35-s − 1.97·37-s + 0.973·38-s + 1.42·40-s + 0.312·41-s + 0.152·43-s − 0.884·46-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

\(L(1)\) \(\approx\) \(1.904333576\)
\(L(\frac12)\) \(\approx\) \(1.904333576\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.92606092903568, −12.29253811942827, −12.23938635834460, −11.75268584113811, −11.26168357887548, −10.81154665131581, −9.980708569634507, −9.856056692645426, −9.132366449078351, −8.661987081017386, −8.168427915057969, −7.887643654720180, −7.243907532897524, −6.701138209664372, −6.171204906616953, −5.633486424327431, −5.143242863724325, −4.617829549406465, −4.039538946402450, −3.555102456116167, −3.420249151469354, −2.738704657744038, −1.854679364647363, −0.8907897708873397, −0.4461199375392742, 0.4461199375392742, 0.8907897708873397, 1.854679364647363, 2.738704657744038, 3.420249151469354, 3.555102456116167, 4.039538946402450, 4.617829549406465, 5.143242863724325, 5.633486424327431, 6.171204906616953, 6.701138209664372, 7.243907532897524, 7.887643654720180, 8.168427915057969, 8.661987081017386, 9.132366449078351, 9.856056692645426, 9.980708569634507, 10.81154665131581, 11.26168357887548, 11.75268584113811, 12.23938635834460, 12.29253811942827, 12.92606092903568

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