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Properties

Label 2-21-7.6-c6-0-3
Degree $2$
Conductor $21$
Sign $0.967 - 0.252i$
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  + 13.8·2-s + 15.5i·3-s + 128.·4-s − 109. i·5-s + 216. i·6-s + (86.6 + 331. i)7-s + 898.·8-s − 243·9-s − 1.51e3i·10-s − 1.73e3·11-s + 2.00e3i·12-s − 3.26e3i·13-s + (1.20e3 + 4.60e3i)14-s + 1.70e3·15-s + 4.23e3·16-s − 1.69e3i·17-s + ⋯
L(s)  = 1  + 1.73·2-s + 0.577i·3-s + 2.01·4-s − 0.873i·5-s + 1.00i·6-s + (0.252 + 0.967i)7-s + 1.75·8-s − 0.333·9-s − 1.51i·10-s − 1.30·11-s + 1.16i·12-s − 1.48i·13-s + (0.438 + 1.67i)14-s + 0.504·15-s + 1.03·16-s − 0.345i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.967 - 0.252i$
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: \((2,\ 21,\ (\ :3),\ 0.967 - 0.252i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.54223 + 0.454947i\)
\(L(\frac12)\) \(\approx\) \(3.54223 + 0.454947i\)
\(L(4)\) 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 \( 1 + (-86.6 - 331. i)T \)
good2 \( 1 - 13.8T + 64T^{2} \)
5 \( 1 + 109. iT - 1.56e4T^{2} \)
11 \( 1 + 1.73e3T + 1.77e6T^{2} \)
13 \( 1 + 3.26e3iT - 4.82e6T^{2} \)
17 \( 1 + 1.69e3iT - 2.41e7T^{2} \)
19 \( 1 - 5.84e3iT - 4.70e7T^{2} \)
23 \( 1 + 9.18e3T + 1.48e8T^{2} \)
29 \( 1 - 3.92e4T + 5.94e8T^{2} \)
31 \( 1 - 5.26e4iT - 8.87e8T^{2} \)
37 \( 1 - 3.18e4T + 2.56e9T^{2} \)
41 \( 1 + 3.03e4iT - 4.75e9T^{2} \)
43 \( 1 - 4.93e4T + 6.32e9T^{2} \)
47 \( 1 + 5.72e3iT - 1.07e10T^{2} \)
53 \( 1 - 9.58e4T + 2.21e10T^{2} \)
59 \( 1 + 2.67e5iT - 4.21e10T^{2} \)
61 \( 1 + 8.35e4iT - 5.15e10T^{2} \)
67 \( 1 + 4.52e5T + 9.04e10T^{2} \)
71 \( 1 - 5.52e4T + 1.28e11T^{2} \)
73 \( 1 - 5.75e5iT - 1.51e11T^{2} \)
79 \( 1 + 5.47e4T + 2.43e11T^{2} \)
83 \( 1 - 2.04e5iT - 3.26e11T^{2} \)
89 \( 1 + 7.48e5iT - 4.96e11T^{2} \)
97 \( 1 + 4.24e5iT - 8.32e11T^{2} \)
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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

−16.02928344690506877423207467599, −15.58480154763716729198630376024, −14.29495518863395506647882750501, −12.85940210104741385199286410199, −12.14976324593196939689370691007, −10.47834950288066901215559326368, −8.270341877813840240814173644742, −5.65781257829414900083514175011, −4.89056827943227621558615405797, −2.88090224839892098396439215110, 2.54431997975118844648426017328, 4.40859184015127992452205912707, 6.34139075785669813822790860995, 7.45449901057765379949122788002, 10.65872990082252812104275026340, 11.73363288291533033833917710728, 13.26374012307886931009786093859, 13.93483614666225766178586067504, 14.99041126495885583273692256796, 16.38740170094574734479017282638

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