Properties

Label 2-221067-1.1-c1-0-8
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 + 4·16-s + 3·17-s + 6·19-s − 2·20-s + 8·23-s − 4·25-s + 2·28-s + 29-s + 6·31-s − 35-s − 2·37-s − 6·41-s + 43-s − 3·47-s + 49-s − 12·53-s − 7·59-s + 12·61-s − 8·64-s − 11·67-s − 6·68-s − 8·71-s − 4·73-s − 12·76-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 0.377·7-s + 16-s + 0.727·17-s + 1.37·19-s − 0.447·20-s + 1.66·23-s − 4/5·25-s + 0.377·28-s + 0.185·29-s + 1.07·31-s − 0.169·35-s − 0.328·37-s − 0.937·41-s + 0.152·43-s − 0.437·47-s + 1/7·49-s − 1.64·53-s − 0.911·59-s + 1.53·61-s − 64-s − 1.34·67-s − 0.727·68-s − 0.949·71-s − 0.468·73-s − 1.37·76-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

\(L(1)\) \(\approx\) \(1.672654939\)
\(L(\frac12)\) \(\approx\) \(1.672654939\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
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

−13.18509951203728, −12.54014989435144, −12.04842133074260, −11.75072299314651, −11.04554330636122, −10.53341407297995, −9.967507048166699, −9.648153049754152, −9.404843111665107, −8.729373450157993, −8.376473763053181, −7.745989271143388, −7.341599461871784, −6.763148370518319, −6.106548261816406, −5.720529798445431, −5.027090712156039, −4.938435261279816, −4.163534747687304, −3.496478108045201, −3.093767372124982, −2.660250139223807, −1.470958370903904, −1.253890140657417, −0.3943194574524129, 0.3943194574524129, 1.253890140657417, 1.470958370903904, 2.660250139223807, 3.093767372124982, 3.496478108045201, 4.163534747687304, 4.938435261279816, 5.027090712156039, 5.720529798445431, 6.106548261816406, 6.763148370518319, 7.341599461871784, 7.745989271143388, 8.376473763053181, 8.729373450157993, 9.404843111665107, 9.648153049754152, 9.967507048166699, 10.53341407297995, 11.04554330636122, 11.75072299314651, 12.04842133074260, 12.54014989435144, 13.18509951203728

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