Properties

Label 2-21-1.1-c1-0-0
Degree $2$
Conductor $21$
Sign $1$
Analytic cond. $0.167685$
Root an. cond. $0.409494$
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 + 3-s − 4-s − 2·5-s − 6-s − 7-s + 3·8-s + 9-s + 2·10-s + 4·11-s − 12-s − 2·13-s + 14-s − 2·15-s − 16-s − 6·17-s − 18-s + 4·19-s + 2·20-s − 21-s − 4·22-s + 3·24-s − 25-s + 2·26-s + 27-s + 28-s − 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.218·21-s − 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.167685\)
Root analytic conductor: \(0.409494\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4511154053\)
\(L(\frac12)\) \(\approx\) \(0.4511154053\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−18.41922826796404611800756026796, −17.15982418199832408435897093924, −15.87345980006494599806967761956, −14.51353234037878627029755764830, −13.19379250527463570089741680145, −11.55595081754559805664353277943, −9.724661210726173973340366981146, −8.672365196683144779961098572501, −7.25047783802838427330028450541, −4.13559084050773741974089362056, 4.13559084050773741974089362056, 7.25047783802838427330028450541, 8.672365196683144779961098572501, 9.724661210726173973340366981146, 11.55595081754559805664353277943, 13.19379250527463570089741680145, 14.51353234037878627029755764830, 15.87345980006494599806967761956, 17.15982418199832408435897093924, 18.41922826796404611800756026796

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