Properties

Label 2-221067-1.1-c1-0-20
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 − 6·19-s + 2·20-s − 4·23-s − 25-s − 2·26-s − 28-s − 29-s + 4·31-s − 5·32-s − 2·34-s − 2·35-s + 6·37-s + 6·38-s − 6·40-s + 2·41-s + 12·43-s + 4·46-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 − 1.37·19-s + 0.447·20-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.185·29-s + 0.718·31-s − 0.883·32-s − 0.342·34-s − 0.338·35-s + 0.986·37-s + 0.973·38-s − 0.948·40-s + 0.312·41-s + 1.82·43-s + 0.589·46-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.06163481330351, −12.70905736809367, −12.31018757537253, −11.72216724914513, −11.13800267481172, −10.96220660935412, −10.32109200550328, −9.982394676676122, −9.328093695657272, −8.957110770378146, −8.417189853957026, −7.990976593287798, −7.792949333508615, −7.250197185478705, −6.590666327034465, −5.898668753872392, −5.648322996190665, −4.663385404020040, −4.393734261183900, −3.980202449538226, −3.496637814348642, −2.589693043151882, −2.036061386490724, −1.252161397215463, −0.6950854682080224, 0, 0.6950854682080224, 1.252161397215463, 2.036061386490724, 2.589693043151882, 3.496637814348642, 3.980202449538226, 4.393734261183900, 4.663385404020040, 5.648322996190665, 5.898668753872392, 6.590666327034465, 7.250197185478705, 7.792949333508615, 7.990976593287798, 8.417189853957026, 8.957110770378146, 9.328093695657272, 9.982394676676122, 10.32109200550328, 10.96220660935412, 11.13800267481172, 11.72216724914513, 12.31018757537253, 12.70905736809367, 13.06163481330351

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