Properties

Label 2-221067-1.1-c1-0-23
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 − 5-s + 7-s + 3·8-s + 10-s + 13-s − 14-s − 16-s + 5·17-s + 2·19-s + 20-s − 6·23-s − 4·25-s − 26-s − 28-s + 29-s − 3·31-s − 5·32-s − 5·34-s − 35-s − 8·37-s − 2·38-s − 3·40-s + 2·41-s + 11·43-s + 6·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.316·10-s + 0.277·13-s − 0.267·14-s − 1/4·16-s + 1.21·17-s + 0.458·19-s + 0.223·20-s − 1.25·23-s − 4/5·25-s − 0.196·26-s − 0.188·28-s + 0.185·29-s − 0.538·31-s − 0.883·32-s − 0.857·34-s − 0.169·35-s − 1.31·37-s − 0.324·38-s − 0.474·40-s + 0.312·41-s + 1.67·43-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: \(1\)
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 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−13.26723288540344, −12.55165353620442, −12.27491252836319, −11.82287042699709, −11.27342391347807, −10.74088569560549, −10.36811222514042, −9.860846178347861, −9.464689359995244, −8.987031510724394, −8.421426600696963, −8.037715228065255, −7.622241767018431, −7.332198701820197, −6.633290551672423, −5.772468477286451, −5.599778213535188, −4.958076657653377, −4.340963201835944, −3.752335518250052, −3.606135172007548, −2.621521059081335, −1.933673303232036, −1.328320783134166, −0.7355496663609853, 0, 0.7355496663609853, 1.328320783134166, 1.933673303232036, 2.621521059081335, 3.606135172007548, 3.752335518250052, 4.340963201835944, 4.958076657653377, 5.599778213535188, 5.772468477286451, 6.633290551672423, 7.332198701820197, 7.622241767018431, 8.037715228065255, 8.421426600696963, 8.987031510724394, 9.464689359995244, 9.860846178347861, 10.36811222514042, 10.74088569560549, 11.27342391347807, 11.82287042699709, 12.27491252836319, 12.55165353620442, 13.26723288540344

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