Properties

Label 2-221067-1.1-c1-0-4
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 − 5-s − 7-s + 4·13-s + 4·16-s + 17-s − 6·19-s + 2·20-s − 6·23-s − 4·25-s + 2·28-s + 29-s + 4·31-s + 35-s − 6·37-s + 6·41-s + 5·43-s − 7·47-s + 49-s − 8·52-s + 4·53-s + 11·59-s + 8·61-s − 8·64-s − 4·65-s + 67-s − 2·68-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s − 0.377·7-s + 1.10·13-s + 16-s + 0.242·17-s − 1.37·19-s + 0.447·20-s − 1.25·23-s − 4/5·25-s + 0.377·28-s + 0.185·29-s + 0.718·31-s + 0.169·35-s − 0.986·37-s + 0.937·41-s + 0.762·43-s − 1.02·47-s + 1/7·49-s − 1.10·52-s + 0.549·53-s + 1.43·59-s + 1.02·61-s − 64-s − 0.496·65-s + 0.122·67-s − 0.242·68-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

\(L(1)\) \(\approx\) \(0.9471182972\)
\(L(\frac12)\) \(\approx\) \(0.9471182972\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.07495204345577, −12.42153168408561, −12.20105328457794, −11.67845340900005, −10.93165673942756, −10.72790207799277, −10.06833061010079, −9.668278457813231, −9.299311562798085, −8.453606505474923, −8.356934355833629, −8.066720391137851, −7.296356000108568, −6.633771209916208, −6.284132632897838, −5.596757988718451, −5.385058184037105, −4.429730598716783, −4.048450268518349, −3.859706938179202, −3.191930037823334, −2.434211428885868, −1.773111099695980, −0.9756434596537157, −0.3273156854736265, 0.3273156854736265, 0.9756434596537157, 1.773111099695980, 2.434211428885868, 3.191930037823334, 3.859706938179202, 4.048450268518349, 4.429730598716783, 5.385058184037105, 5.596757988718451, 6.284132632897838, 6.633771209916208, 7.296356000108568, 8.066720391137851, 8.356934355833629, 8.453606505474923, 9.299311562798085, 9.668278457813231, 10.06833061010079, 10.72790207799277, 10.93165673942756, 11.67845340900005, 12.20105328457794, 12.42153168408561, 13.07495204345577

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