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Properties

Label 2-21-7.6-c6-0-2
Degree $2$
Conductor $21$
Sign $-0.0132 - 0.999i$
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  + 5.57·2-s + 15.5i·3-s − 32.9·4-s + 205. i·5-s + 86.8i·6-s + (342. − 4.53i)7-s − 540.·8-s − 243·9-s + 1.14e3i·10-s + 1.02e3·11-s − 513. i·12-s + 281. i·13-s + (1.91e3 − 25.2i)14-s − 3.20e3·15-s − 901.·16-s + 1.17e3i·17-s + ⋯
L(s)  = 1  + 0.696·2-s + 0.577i·3-s − 0.514·4-s + 1.64i·5-s + 0.402i·6-s + (0.999 − 0.0132i)7-s − 1.05·8-s − 0.333·9-s + 1.14i·10-s + 0.769·11-s − 0.297i·12-s + 0.128i·13-s + (0.696 − 0.00921i)14-s − 0.949·15-s − 0.220·16-s + 0.239i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0132 - 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0132 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.0132 - 0.999i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ -0.0132 - 0.999i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.30365 + 1.32101i\)
\(L(\frac12)\) \(\approx\) \(1.30365 + 1.32101i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 15.5iT \)
7 \( 1 + (-342. + 4.53i)T \)
good2 \( 1 - 5.57T + 64T^{2} \)
5 \( 1 - 205. iT - 1.56e4T^{2} \)
11 \( 1 - 1.02e3T + 1.77e6T^{2} \)
13 \( 1 - 281. iT - 4.82e6T^{2} \)
17 \( 1 - 1.17e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.02e4iT - 4.70e7T^{2} \)
23 \( 1 - 1.81e4T + 1.48e8T^{2} \)
29 \( 1 - 1.46e4T + 5.94e8T^{2} \)
31 \( 1 - 5.26e4iT - 8.87e8T^{2} \)
37 \( 1 + 7.82e3T + 2.56e9T^{2} \)
41 \( 1 - 3.52e4iT - 4.75e9T^{2} \)
43 \( 1 + 4.56e4T + 6.32e9T^{2} \)
47 \( 1 + 1.58e5iT - 1.07e10T^{2} \)
53 \( 1 + 4.01e3T + 2.21e10T^{2} \)
59 \( 1 - 1.69e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.98e5iT - 5.15e10T^{2} \)
67 \( 1 + 5.25e5T + 9.04e10T^{2} \)
71 \( 1 - 2.82e4T + 1.28e11T^{2} \)
73 \( 1 + 5.84e5iT - 1.51e11T^{2} \)
79 \( 1 - 3.39e5T + 2.43e11T^{2} \)
83 \( 1 - 1.34e5iT - 3.26e11T^{2} \)
89 \( 1 + 9.03e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.07e6iT - 8.32e11T^{2} \)
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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

−17.38884622395751175593222986131, −15.21214933754913186938814132513, −14.63845661632557763863151380557, −13.70503986365312666206003299130, −11.68436258946687148022581195183, −10.61985167663038750895511250855, −8.912745215395188582788199581482, −6.76209360574767100770578176833, −4.84315718546711764132361407739, −3.18885563788264528961482871701, 1.12012393647685638525775086017, 4.36555836651877663133767365553, 5.60439357554383956676665846764, 8.137925979537713895160211294052, 9.196806393800549538175229467761, 11.78797688203601549355179843598, 12.65902274928321199134127789388, 13.72127956550613465848775516134, 14.86950333760972562286949097072, 16.73945818169898523174728461481

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