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Properties

Label 2-21-21.20-c39-0-41
Degree $2$
Conductor $21$
Sign $0.425 - 0.905i$
Analytic cond. $202.313$
Root an. cond. $14.2236$
Motivic weight $39$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.01e9i·3-s + 5.49e11·4-s + (−2.72e16 − 1.28e16i)7-s − 4.05e18·9-s + 1.10e21i·12-s + 3.71e20i·13-s + 3.02e23·16-s − 3.91e24i·19-s + (2.58e25 − 5.49e25i)21-s − 1.81e27·25-s − 8.15e27i·27-s + (−1.50e28 − 7.04e27i)28-s − 4.61e28i·31-s − 2.22e30·36-s + 5.15e29·37-s + ⋯
L(s)  = 1  + 1.00i·3-s + 4-s + (−0.905 − 0.425i)7-s − 1.00·9-s + 1.00i·12-s + 0.0704i·13-s + 16-s − 0.453i·19-s + (0.425 − 0.905i)21-s − 25-s − 1.00i·27-s + (−0.905 − 0.425i)28-s − 0.382i·31-s − 1.00·36-s + 0.135·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.425 - 0.905i$
Analytic conductor: \(202.313\)
Root analytic conductor: \(14.2236\)
Motivic weight: \(39\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :39/2),\ 0.425 - 0.905i)\)

Particular Values

\(L(20)\) \(\approx\) \(2.228133614\)
\(L(\frac12)\) \(\approx\) \(2.228133614\)
\(L(\frac{41}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 2.01e9iT \)
7 \( 1 + (2.72e16 + 1.28e16i)T \)
good2 \( 1 - 5.49e11T^{2} \)
5 \( 1 + 1.81e27T^{2} \)
11 \( 1 - 4.11e40T^{2} \)
13 \( 1 - 3.71e20iT - 2.77e43T^{2} \)
17 \( 1 + 9.71e47T^{2} \)
19 \( 1 + 3.91e24iT - 7.43e49T^{2} \)
23 \( 1 - 1.28e53T^{2} \)
29 \( 1 - 1.08e57T^{2} \)
31 \( 1 + 4.61e28iT - 1.45e58T^{2} \)
37 \( 1 - 5.15e29T + 1.44e61T^{2} \)
41 \( 1 + 7.91e62T^{2} \)
43 \( 1 - 6.23e31T + 5.07e63T^{2} \)
47 \( 1 + 1.62e65T^{2} \)
53 \( 1 - 1.76e67T^{2} \)
59 \( 1 + 1.15e69T^{2} \)
61 \( 1 - 1.04e35iT - 4.24e69T^{2} \)
67 \( 1 - 3.61e35T + 1.64e71T^{2} \)
71 \( 1 - 1.58e72T^{2} \)
73 \( 1 + 3.15e36iT - 4.67e72T^{2} \)
79 \( 1 - 1.09e37T + 1.01e74T^{2} \)
83 \( 1 + 6.98e74T^{2} \)
89 \( 1 + 1.06e76T^{2} \)
97 \( 1 - 9.21e38iT - 3.04e77T^{2} \)
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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

−11.06141554185866657944236902544, −10.18941107291136916475295904162, −9.248608114800294806487062099725, −7.75415805842596426854711322157, −6.54515711977350486475237043970, −5.61255075580973117695747334206, −4.15090282613603660968882864534, −3.21990280785127080963323779897, −2.26873826374979445380712122208, −0.68474147091730658447151654117, 0.52858109233238948700668611307, 1.72030802078574462773256295154, 2.55958447963267854874124540278, 3.51476135388203516136593097706, 5.60082413014624575458279490187, 6.35020690633978264816665195188, 7.24996454380325352572065411446, 8.293243868561784737823190532676, 9.724489876390417096211203571782, 11.08032216004088183742248738201

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