Properties

Label 2-221067-1.1-c1-0-24
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 $1$

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 − 17-s + 3·19-s − 2·20-s + 23-s − 25-s + 2·26-s + 28-s − 29-s + 7·31-s − 5·32-s + 34-s − 2·35-s − 3·38-s + 6·40-s + 5·41-s − 6·43-s − 46-s − 11·47-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 − 0.242·17-s + 0.688·19-s − 0.447·20-s + 0.208·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.185·29-s + 1.25·31-s − 0.883·32-s + 0.171·34-s − 0.338·35-s − 0.486·38-s + 0.948·40-s + 0.780·41-s − 0.914·43-s − 0.147·46-s − 1.60·47-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: \(1\)
Selberg data: \((2,\ 221067,\ (\ :1/2),\ -1)\)

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)$
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 \( 1 + T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.27938483719600, −12.89511807581979, −12.25165011975208, −11.79538432915626, −11.27707896781450, −10.66599033377273, −10.13379939981072, −9.924752941603219, −9.460329305103947, −9.158300939640133, −8.529186205147309, −8.105213368909359, −7.619353497623598, −7.011545984085714, −6.626365275311666, −5.976204828562723, −5.462783801051535, −5.017551558821482, −4.466291382358210, −3.935958177389776, −3.183931251146194, −2.659846892763831, −1.969400439285055, −1.415575751406481, −0.7444067419362038, 0, 0.7444067419362038, 1.415575751406481, 1.969400439285055, 2.659846892763831, 3.183931251146194, 3.935958177389776, 4.466291382358210, 5.017551558821482, 5.462783801051535, 5.976204828562723, 6.626365275311666, 7.011545984085714, 7.619353497623598, 8.105213368909359, 8.529186205147309, 9.158300939640133, 9.460329305103947, 9.924752941603219, 10.13379939981072, 10.66599033377273, 11.27707896781450, 11.79538432915626, 12.25165011975208, 12.89511807581979, 13.27938483719600

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