Properties

Label 2-21-21.11-c6-0-13
Degree $2$
Conductor $21$
Sign $-0.348 + 0.937i$
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  + (13.1 − 7.60i)2-s + (−23.5 − 13.1i)3-s + (83.5 − 144. i)4-s + (−82.1 + 47.4i)5-s + (−410. + 5.43i)6-s + (334. − 73.9i)7-s − 1.56e3i·8-s + (381. + 621. i)9-s + (−721. + 1.24e3i)10-s + (735. + 424. i)11-s + (−3.87e3 + 2.30e3i)12-s + 1.14e3·13-s + (3.84e3 − 3.51e3i)14-s + (2.56e3 − 33.9i)15-s + (−6.56e3 − 1.13e4i)16-s + (−2.38e3 − 1.37e3i)17-s + ⋯
L(s)  = 1  + (1.64 − 0.950i)2-s + (−0.872 − 0.488i)3-s + (1.30 − 2.26i)4-s + (−0.657 + 0.379i)5-s + (−1.90 + 0.0251i)6-s + (0.976 − 0.215i)7-s − 3.06i·8-s + (0.522 + 0.852i)9-s + (−0.721 + 1.24i)10-s + (0.552 + 0.319i)11-s + (−2.24 + 1.33i)12-s + 0.520·13-s + (1.40 − 1.28i)14-s + (0.758 − 0.0100i)15-s + (−1.60 − 2.77i)16-s + (−0.484 − 0.279i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.348 + 0.937i$
Analytic conductor: \(4.83113\)
Root analytic conductor: \(2.19798\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3),\ -0.348 + 0.937i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.58664 - 2.28396i\)
\(L(\frac12)\) \(\approx\) \(1.58664 - 2.28396i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (23.5 + 13.1i)T \)
7 \( 1 + (-334. + 73.9i)T \)
good2 \( 1 + (-13.1 + 7.60i)T + (32 - 55.4i)T^{2} \)
5 \( 1 + (82.1 - 47.4i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-735. - 424. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 1.14e3T + 4.82e6T^{2} \)
17 \( 1 + (2.38e3 + 1.37e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-3.55e3 - 6.15e3i)T + (-2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (4.81e3 - 2.78e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 2.27e4iT - 5.94e8T^{2} \)
31 \( 1 + (4.26e3 - 7.37e3i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (3.59e4 + 6.22e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 5.90e4iT - 4.75e9T^{2} \)
43 \( 1 + 3.77e4T + 6.32e9T^{2} \)
47 \( 1 + (9.76e4 - 5.63e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-1.75e5 - 1.01e5i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-1.50e5 - 8.66e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-2.45e4 - 4.25e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.56e5 - 2.71e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 1.00e5iT - 1.28e11T^{2} \)
73 \( 1 + (-1.41e5 + 2.45e5i)T + (-7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (1.86e5 + 3.22e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 2.43e5iT - 3.26e11T^{2} \)
89 \( 1 + (-8.90e5 + 5.13e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.64e6T + 8.32e11T^{2} \)
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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

−16.04443988806482485598600693540, −14.71271503350349739871816376130, −13.65122425013430309964114359133, −12.23510218605903045336164277077, −11.48116769835021597038554998188, −10.59747318027983047639021294009, −7.12288290249451917432769794651, −5.47637981511309625055834103026, −4.00532660586457944601758641856, −1.56126063629787828727149570808, 4.00881389920252961136597994153, 5.08490965104130443105968697511, 6.52381329258073545136257189685, 8.265053829008698900252789065630, 11.37907781583403601552140564318, 11.94993596819171100742600808191, 13.46898013227059878653523928080, 14.91811565228472757445771832401, 15.71523704658554733290475541588, 16.69254190315937193320886942772

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