Properties

Label 2-21-7.6-c6-0-5
Degree $2$
Conductor $21$
Sign $-0.807 - 0.589i$
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  − 14.3·2-s − 15.5i·3-s + 141.·4-s − 175. i·5-s + 223. i·6-s + (−202. + 276. i)7-s − 1.10e3·8-s − 243·9-s + 2.51e3i·10-s − 998.·11-s − 2.20e3i·12-s + 201. i·13-s + (2.89e3 − 3.96e3i)14-s − 2.73e3·15-s + 6.80e3·16-s + 8.77e3i·17-s + ⋯
L(s)  = 1  − 1.79·2-s − 0.577i·3-s + 2.20·4-s − 1.40i·5-s + 1.03i·6-s + (−0.589 + 0.807i)7-s − 2.16·8-s − 0.333·9-s + 2.51i·10-s − 0.750·11-s − 1.27i·12-s + 0.0916i·13-s + (1.05 − 1.44i)14-s − 0.810·15-s + 1.66·16-s + 1.78i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.807 - 0.589i$
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: \((2,\ 21,\ (\ :3),\ -0.807 - 0.589i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0184581 + 0.0565473i\)
\(L(\frac12)\) \(\approx\) \(0.0184581 + 0.0565473i\)
\(L(4)\) 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 \( 1 + (202. - 276. i)T \)
good2 \( 1 + 14.3T + 64T^{2} \)
5 \( 1 + 175. iT - 1.56e4T^{2} \)
11 \( 1 + 998.T + 1.77e6T^{2} \)
13 \( 1 - 201. iT - 4.82e6T^{2} \)
17 \( 1 - 8.77e3iT - 2.41e7T^{2} \)
19 \( 1 + 1.68e3iT - 4.70e7T^{2} \)
23 \( 1 + 2.15e4T + 1.48e8T^{2} \)
29 \( 1 - 9.51e3T + 5.94e8T^{2} \)
31 \( 1 + 1.62e4iT - 8.87e8T^{2} \)
37 \( 1 - 1.04e4T + 2.56e9T^{2} \)
41 \( 1 + 5.79e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.04e5T + 6.32e9T^{2} \)
47 \( 1 + 3.39e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.20e5T + 2.21e10T^{2} \)
59 \( 1 - 2.56e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.21e5iT - 5.15e10T^{2} \)
67 \( 1 - 2.16e4T + 9.04e10T^{2} \)
71 \( 1 + 9.83e4T + 1.28e11T^{2} \)
73 \( 1 + 1.44e5iT - 1.51e11T^{2} \)
79 \( 1 + 5.07e5T + 2.43e11T^{2} \)
83 \( 1 + 5.54e5iT - 3.26e11T^{2} \)
89 \( 1 + 9.94e5iT - 4.96e11T^{2} \)
97 \( 1 - 9.61e5iT - 8.32e11T^{2} \)
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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

−16.45947182866522950617209943979, −15.51894385330833743617315239522, −12.94581526196057041303789201887, −11.94441742015507934011583944598, −10.12971061016744390194473032538, −8.789991353665559449908321845253, −8.037793195964701898244432620613, −6.05147874704189034122027293559, −1.86031590699950745970663725554, −0.06203558160706352611186917070, 2.88332618588546270356124984421, 6.65723166895096874691935264565, 7.81249051765141522332693924587, 9.759242117835276051135127856400, 10.35521252837817604756438314650, 11.45276275449627947459111816127, 14.08051006880008763702202392004, 15.67340694944786469140107368089, 16.40202825337606961077498949920, 17.88552957448676300315604806451

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