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Properties

Label 2-21-21.20-c33-0-9
Degree $2$
Conductor $21$
Sign $-0.999 + 0.00904i$
Analytic cond. $144.863$
Root an. cond. $12.0359$
Motivic weight $33$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.45e7i·3-s + 8.58e9·4-s + (−7.94e11 − 8.79e13i)7-s − 5.55e15·9-s + 6.40e17i·12-s + 4.74e18i·13-s + 7.37e19·16-s − 1.99e21i·19-s + (6.55e21 − 5.92e19i)21-s − 1.16e23·25-s − 4.14e23i·27-s + (−6.82e21 − 7.55e23i)28-s + 7.32e24i·31-s − 4.77e25·36-s + 6.20e25·37-s + ⋯
L(s)  = 1  + 0.999i·3-s + 4-s + (−0.00904 − 0.999i)7-s − 9-s + 0.999i·12-s + 1.97i·13-s + 16-s − 1.58i·19-s + (0.999 − 0.00904i)21-s − 25-s − 1.00i·27-s + (−0.00904 − 0.999i)28-s + 1.80i·31-s − 36-s + 0.826·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00904i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.999 + 0.00904i$
Analytic conductor: \(144.863\)
Root analytic conductor: \(12.0359\)
Motivic weight: \(33\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :33/2),\ -0.999 + 0.00904i)\)

Particular Values

\(L(17)\) \(\approx\) \(1.113859851\)
\(L(\frac12)\) \(\approx\) \(1.113859851\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 7.45e7iT \)
7 \( 1 + (7.94e11 + 8.79e13i)T \)
good2 \( 1 - 8.58e9T^{2} \)
5 \( 1 + 1.16e23T^{2} \)
11 \( 1 - 2.32e34T^{2} \)
13 \( 1 - 4.74e18iT - 5.75e36T^{2} \)
17 \( 1 + 4.02e40T^{2} \)
19 \( 1 + 1.99e21iT - 1.58e42T^{2} \)
23 \( 1 - 8.65e44T^{2} \)
29 \( 1 - 1.81e48T^{2} \)
31 \( 1 - 7.32e24iT - 1.64e49T^{2} \)
37 \( 1 - 6.20e25T + 5.63e51T^{2} \)
41 \( 1 + 1.66e53T^{2} \)
43 \( 1 + 5.99e26T + 8.02e53T^{2} \)
47 \( 1 + 1.51e55T^{2} \)
53 \( 1 - 7.96e56T^{2} \)
59 \( 1 + 2.74e58T^{2} \)
61 \( 1 - 2.96e29iT - 8.23e58T^{2} \)
67 \( 1 + 1.96e30T + 1.82e60T^{2} \)
71 \( 1 - 1.23e61T^{2} \)
73 \( 1 - 2.30e30iT - 3.08e61T^{2} \)
79 \( 1 + 3.82e31T + 4.18e62T^{2} \)
83 \( 1 + 2.13e63T^{2} \)
89 \( 1 + 2.13e64T^{2} \)
97 \( 1 - 6.98e32iT - 3.65e65T^{2} \)
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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

−11.75597968710494171526101057403, −11.08355023186861475243998315062, −10.03463981133916323379650735865, −8.882101089519224247129425864881, −7.26992270578374018833452595133, −6.39072102657627412142429426524, −4.80337174425288640686307785829, −3.85022615922826432676959765542, −2.64432656460013503773393470703, −1.37191212566506567442136201734, 0.18965131431482006915860453698, 1.47879186185359150905467252630, 2.40186040588091505622487788367, 3.28788307723655027109184455758, 5.70823230109624286063943868437, 6.01991648491591392154873020392, 7.64329256812074160581944276351, 8.184144687933416395144613324777, 10.00791387526172531542949872125, 11.36319604855647641634955305864

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