Properties

Label 2-21-7.3-c6-0-2
Degree $2$
Conductor $21$
Sign $-0.984 - 0.178i$
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  + (5.80 + 10.0i)2-s + (−13.5 − 7.79i)3-s + (−35.4 + 61.4i)4-s + (−165. + 95.4i)5-s − 181. i·6-s + (−103. + 327. i)7-s − 80.4·8-s + (121.5 + 210. i)9-s + (−1.92e3 − 1.10e3i)10-s + (1.02e3 − 1.77e3i)11-s + (957. − 552. i)12-s + 3.05e3i·13-s + (−3.89e3 + 861. i)14-s + 2.97e3·15-s + (1.80e3 + 3.12e3i)16-s + (2.46e3 + 1.42e3i)17-s + ⋯
L(s)  = 1  + (0.725 + 1.25i)2-s + (−0.5 − 0.288i)3-s + (−0.554 + 0.959i)4-s + (−1.32 + 0.763i)5-s − 0.838i·6-s + (−0.301 + 0.953i)7-s − 0.157·8-s + (0.166 + 0.288i)9-s + (−1.92 − 1.10i)10-s + (0.772 − 1.33i)11-s + (0.554 − 0.319i)12-s + 1.39i·13-s + (−1.41 + 0.313i)14-s + 0.881·15-s + (0.439 + 0.762i)16-s + (0.502 + 0.290i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.178i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.984 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.984 - 0.178i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ -0.984 - 0.178i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.117285 + 1.30676i\)
\(L(\frac12)\) \(\approx\) \(0.117285 + 1.30676i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.5 + 7.79i)T \)
7 \( 1 + (103. - 327. i)T \)
good2 \( 1 + (-5.80 - 10.0i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (165. - 95.4i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-1.02e3 + 1.77e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 3.05e3iT - 4.82e6T^{2} \)
17 \( 1 + (-2.46e3 - 1.42e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (3.42e3 - 1.97e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (330. + 572. i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 9.28e3T + 5.94e8T^{2} \)
31 \( 1 + (-2.42e3 - 1.40e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-1.84e4 - 3.19e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 6.79e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.23e4T + 6.32e9T^{2} \)
47 \( 1 + (-1.27e5 + 7.38e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (1.09e5 - 1.90e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-1.66e5 - 9.62e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-2.88e5 + 1.66e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.74e5 - 3.02e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 3.05e5T + 1.28e11T^{2} \)
73 \( 1 + (2.04e5 + 1.18e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-2.93e5 - 5.07e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 1.06e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.13e4 + 6.52e3i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 2.05e5iT - 8.32e11T^{2} \)
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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

−16.83057316937563270199023615010, −16.04375298198702684754797895112, −14.99403567905021707931893845126, −13.97743308266528590838191023005, −12.21176646723579781760296968359, −11.19937206663194487873251450819, −8.475523699180177711194601600861, −6.97890153111633931684950543511, −5.95182799271489753392232297541, −3.89926635835319500317444055524, 0.74572976032521455474560654063, 3.73960761904721515519785837165, 4.72998357678611418120450307204, 7.54855296551051225826300125306, 9.886359299621615824211730514176, 11.13376880923519923273809705275, 12.26629928900685295506427902189, 12.95255916408358650015389048336, 14.83632242809148568554049467179, 16.20915531880925203816708762557

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