Properties

Label 2-221067-1.1-c1-0-7
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 + 14-s − 16-s + 2·17-s − 2·19-s + 2·20-s − 6·23-s − 25-s − 28-s + 29-s − 8·31-s + 5·32-s + 2·34-s − 2·35-s − 2·37-s − 2·38-s + 6·40-s + 6·41-s − 2·43-s − 6·46-s + 4·47-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 + 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.458·19-s + 0.447·20-s − 1.25·23-s − 1/5·25-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 − 0.328·37-s − 0.324·38-s + 0.948·40-s + 0.937·41-s − 0.304·43-s − 0.884·46-s + 0.583·47-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

\(L(1)\) \(\approx\) \(1.498033647\)
\(L(\frac12)\) \(\approx\) \(1.498033647\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.98557201281870, −12.43055572022303, −12.15184731644882, −11.75849168437586, −11.14526876531540, −10.86225260423123, −10.10880642556753, −9.715575300716065, −9.176919160082507, −8.677306145185549, −8.123100522705120, −7.883064203944979, −7.326615674246064, −6.670561949880390, −6.111704489916586, −5.644239070899118, −5.093957100341908, −4.704699507595723, −4.001843105223707, −3.693199051170661, −3.440908715878732, −2.382581368673546, −2.060688882712044, −0.9934544828409617, −0.3528934216512016, 0.3528934216512016, 0.9934544828409617, 2.060688882712044, 2.382581368673546, 3.440908715878732, 3.693199051170661, 4.001843105223707, 4.704699507595723, 5.093957100341908, 5.644239070899118, 6.111704489916586, 6.670561949880390, 7.326615674246064, 7.883064203944979, 8.123100522705120, 8.677306145185549, 9.176919160082507, 9.715575300716065, 10.10880642556753, 10.86225260423123, 11.14526876531540, 11.75849168437586, 12.15184731644882, 12.43055572022303, 12.98557201281870

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