Properties

Label 2-221-1.1-c1-0-9
Degree $2$
Conductor $221$
Sign $1$
Analytic cond. $1.76469$
Root an. cond. $1.32841$
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 + 2·3-s − 4-s + 2·5-s + 2·6-s + 2·7-s − 3·8-s + 9-s + 2·10-s − 6·11-s − 2·12-s − 13-s + 2·14-s + 4·15-s − 16-s + 17-s + 18-s + 4·19-s − 2·20-s + 4·21-s − 6·22-s + 6·23-s − 6·24-s − 25-s − 26-s − 4·27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.894·5-s + 0.816·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 1.80·11-s − 0.577·12-s − 0.277·13-s + 0.534·14-s + 1.03·15-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.872·21-s − 1.27·22-s + 1.25·23-s − 1.22·24-s − 1/5·25-s − 0.196·26-s − 0.769·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $1$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 221,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.176100076\)
\(L(\frac12)\) \(\approx\) \(2.176100076\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−12.81687262876098235036784582433, −11.41508377219727993221954013871, −10.10857183087426188490087859330, −9.297073189927582363453435681927, −8.359798380891594538656347298349, −7.48452778448443093986701738050, −5.57328576129429982536653103768, −5.03216113857164143727970023789, −3.37858390783506699471316087423, −2.31845179494526177837948227739, 2.31845179494526177837948227739, 3.37858390783506699471316087423, 5.03216113857164143727970023789, 5.57328576129429982536653103768, 7.48452778448443093986701738050, 8.359798380891594538656347298349, 9.297073189927582363453435681927, 10.10857183087426188490087859330, 11.41508377219727993221954013871, 12.81687262876098235036784582433

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