Properties

Label 2-221067-1.1-c1-0-27
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 + 2·17-s + 8·19-s − 2·20-s − 25-s + 2·26-s − 28-s + 29-s − 5·32-s − 2·34-s + 2·35-s − 2·37-s − 8·38-s + 6·40-s + 2·41-s − 4·43-s − 8·47-s + 49-s + 50-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 + 0.485·17-s + 1.83·19-s − 0.447·20-s − 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.185·29-s − 0.883·32-s − 0.342·34-s + 0.338·35-s − 0.328·37-s − 1.29·38-s + 0.948·40-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.141·50-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 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.21711252536479, −12.88725695598531, −12.20329491180003, −11.64850796742584, −11.47697549677769, −10.57348807011322, −10.25290948826553, −9.828904380074436, −9.599789523864250, −8.941286170846662, −8.696797622683977, −7.972553556507206, −7.551252807522117, −7.268536998102016, −6.584330580521130, −5.838429976700556, −5.450971678131150, −5.079539173808187, −4.493956614602597, −3.940881401092133, −3.138184308351103, −2.750857461661022, −1.759574187412312, −1.519845614371939, −0.8332110981475053, 0, 0.8332110981475053, 1.519845614371939, 1.759574187412312, 2.750857461661022, 3.138184308351103, 3.940881401092133, 4.493956614602597, 5.079539173808187, 5.450971678131150, 5.838429976700556, 6.584330580521130, 7.268536998102016, 7.551252807522117, 7.972553556507206, 8.696797622683977, 8.941286170846662, 9.599789523864250, 9.828904380074436, 10.25290948826553, 10.57348807011322, 11.47697549677769, 11.64850796742584, 12.20329491180003, 12.88725695598531, 13.21711252536479

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