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Properties

Label 2-21-21.2-c8-0-0
Degree $2$
Conductor $21$
Sign $-0.594 - 0.804i$
Analytic cond. $8.55495$
Root an. cond. $2.92488$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−40.5 − 70.1i)3-s + (−128 − 221. i)4-s + (119.5 + 2.39e3i)7-s + (−3.28e3 + 5.68e3i)9-s + (−1.03e4 + 1.79e4i)12-s − 2.06e4·13-s + (−3.27e4 + 5.67e4i)16-s + (−5.02e4 + 8.70e4i)19-s + (1.63e5 − 1.05e5i)21-s + (−1.95e5 − 3.38e5i)25-s + 5.31e5·27-s + (5.16e5 − 3.33e5i)28-s + (−6.12e5 − 1.06e6i)31-s + 1.67e6·36-s + (−1.48e6 + 2.56e6i)37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.0497 + 0.998i)7-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)12-s − 0.722·13-s + (−0.499 + 0.866i)16-s + (−0.385 + 0.668i)19-s + (0.840 − 0.542i)21-s + (−0.5 − 0.866i)25-s + 27-s + (0.840 − 0.542i)28-s + (−0.663 − 1.14i)31-s + 36-s + (−0.791 + 1.37i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.594 - 0.804i$
Analytic conductor: \(8.55495\)
Root analytic conductor: \(2.92488\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :4),\ -0.594 - 0.804i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0122476 + 0.0242905i\)
\(L(\frac12)\) \(\approx\) \(0.0122476 + 0.0242905i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (40.5 + 70.1i)T \)
7 \( 1 + (-119.5 - 2.39e3i)T \)
good2 \( 1 + (128 + 221. i)T^{2} \)
5 \( 1 + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + 2.06e4T + 8.15e8T^{2} \)
17 \( 1 + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (5.02e4 - 8.70e4i)T + (-8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 - 5.00e11T^{2} \)
31 \( 1 + (6.12e5 + 1.06e6i)T + (-4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (1.48e6 - 2.56e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 7.98e12T^{2} \)
43 \( 1 + 6.83e6T + 1.16e13T^{2} \)
47 \( 1 + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-1.19e7 + 2.06e7i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (1.86e7 + 3.22e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 6.45e14T^{2} \)
73 \( 1 + (-2.76e7 - 4.78e7i)T + (-4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (3.74e7 - 6.48e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 2.25e15T^{2} \)
89 \( 1 + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 1.76e8T + 7.83e15T^{2} \)
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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

−16.98491894259675050641339044290, −15.29200932725639350088008541906, −14.11315272123204943681388978184, −12.76894787036919711784978451655, −11.57239418173787500276244334616, −9.954580578670223570415162559704, −8.313267813178255265750901290622, −6.34768266429729243552035725998, −5.13873438574477884754289261089, −1.93347095642437217861327855793, 0.01416689557175330012724977904, 3.58138282006602117169220338107, 4.89786215999606783991360831706, 7.20321873157896985087515750853, 8.979395043871960646558849491891, 10.36265179348038408059423182347, 11.75080971989746328731517973111, 13.17442605383881753167377198218, 14.58954303639513950325049632208, 16.15264425770588581663712642672

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