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Properties

Label 2-21-3.2-c6-0-3
Degree $2$
Conductor $21$
Sign $-0.173 - 0.984i$
Analytic cond. $4.83113$
Root an. cond. $2.19798$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.64i·2-s + (−4.69 − 26.5i)3-s + 5.60·4-s + 237. i·5-s + (203. − 35.8i)6-s + 129.·7-s + 531. i·8-s + (−684. + 249. i)9-s − 1.81e3·10-s + 371. i·11-s + (−26.3 − 149. i)12-s + 1.99e3·13-s + 990. i·14-s + (6.30e3 − 1.11e3i)15-s − 3.70e3·16-s − 1.25e3i·17-s + ⋯
L(s)  = 1  + 0.955i·2-s + (−0.173 − 0.984i)3-s + 0.0875·4-s + 1.89i·5-s + (0.940 − 0.166i)6-s + 0.377·7-s + 1.03i·8-s + (−0.939 + 0.342i)9-s − 1.81·10-s + 0.278i·11-s + (−0.0152 − 0.0862i)12-s + 0.909·13-s + 0.361i·14-s + (1.86 − 0.329i)15-s − 0.904·16-s − 0.255i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.00777 + 1.20127i\)
\(L(\frac12)\) \(\approx\) \(1.00777 + 1.20127i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.69 + 26.5i)T \)
7 \( 1 - 129.T \)
good2 \( 1 - 7.64iT - 64T^{2} \)
5 \( 1 - 237. iT - 1.56e4T^{2} \)
11 \( 1 - 371. iT - 1.77e6T^{2} \)
13 \( 1 - 1.99e3T + 4.82e6T^{2} \)
17 \( 1 + 1.25e3iT - 2.41e7T^{2} \)
19 \( 1 - 3.94e3T + 4.70e7T^{2} \)
23 \( 1 + 1.71e4iT - 1.48e8T^{2} \)
29 \( 1 + 2.52e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.86e4T + 8.87e8T^{2} \)
37 \( 1 + 2.91e4T + 2.56e9T^{2} \)
41 \( 1 + 3.97e3iT - 4.75e9T^{2} \)
43 \( 1 - 8.45e4T + 6.32e9T^{2} \)
47 \( 1 - 7.31e4iT - 1.07e10T^{2} \)
53 \( 1 - 4.05e4iT - 2.21e10T^{2} \)
59 \( 1 + 3.77e4iT - 4.21e10T^{2} \)
61 \( 1 - 4.03e5T + 5.15e10T^{2} \)
67 \( 1 + 1.92e3T + 9.04e10T^{2} \)
71 \( 1 + 3.37e5iT - 1.28e11T^{2} \)
73 \( 1 - 1.69e5T + 1.51e11T^{2} \)
79 \( 1 + 1.58e5T + 2.43e11T^{2} \)
83 \( 1 - 3.25e4iT - 3.26e11T^{2} \)
89 \( 1 + 1.05e6iT - 4.96e11T^{2} \)
97 \( 1 + 7.48e5T + 8.32e11T^{2} \)
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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

−17.41480313077268673116877615114, −15.75458659361740312274595614724, −14.57167814178738082788734060719, −13.84148592886703285408713763643, −11.69930894636366959380176776677, −10.74479946051779223277540307913, −8.026872579420880627941352977636, −6.94978241246093323675197477074, −6.07689079000269443484871564138, −2.51651143566173053935851017952, 1.13113987237378904863084955544, 3.86643051856995147371921753482, 5.41510531109308055729597557572, 8.554323064297335323905988992893, 9.691959999633636142103950031048, 11.19705132559720158023466312585, 12.17743470409453700878533105843, 13.51706594431538450595551865107, 15.67099504566635870733471432206, 16.30269318592983699709127811748

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