Properties

Label 2-21-21.11-c6-0-0
Degree $2$
Conductor $21$
Sign $-0.683 - 0.729i$
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  + (1.07 − 0.621i)2-s + (−9.77 − 25.1i)3-s + (−31.2 + 54.0i)4-s + (−93.7 + 54.1i)5-s + (−26.1 − 21.0i)6-s + (147. + 309. i)7-s + 157. i·8-s + (−537. + 492. i)9-s + (−67.2 + 116. i)10-s + (−1.89e3 − 1.09e3i)11-s + (1.66e3 + 257. i)12-s − 1.13e3·13-s + (351. + 241. i)14-s + (2.27e3 + 1.83e3i)15-s + (−1.90e3 − 3.29e3i)16-s + (6.94e3 + 4.01e3i)17-s + ⋯
L(s)  = 1  + (0.134 − 0.0776i)2-s + (−0.362 − 0.932i)3-s + (−0.487 + 0.845i)4-s + (−0.749 + 0.433i)5-s + (−0.121 − 0.0972i)6-s + (0.431 + 0.902i)7-s + 0.306i·8-s + (−0.737 + 0.675i)9-s + (−0.0672 + 0.116i)10-s + (−1.42 − 0.822i)11-s + (0.964 + 0.148i)12-s − 0.518·13-s + (0.128 + 0.0878i)14-s + (0.675 + 0.542i)15-s + (−0.464 − 0.803i)16-s + (1.41 + 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.683 - 0.729i$
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: \((2,\ 21,\ (\ :3),\ -0.683 - 0.729i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.173265 + 0.399937i\)
\(L(\frac12)\) \(\approx\) \(0.173265 + 0.399937i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (9.77 + 25.1i)T \)
7 \( 1 + (-147. - 309. i)T \)
good2 \( 1 + (-1.07 + 0.621i)T + (32 - 55.4i)T^{2} \)
5 \( 1 + (93.7 - 54.1i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (1.89e3 + 1.09e3i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 1.13e3T + 4.82e6T^{2} \)
17 \( 1 + (-6.94e3 - 4.01e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (3.23e3 + 5.60e3i)T + (-2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (1.11e3 - 642. i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 1.42e4iT - 5.94e8T^{2} \)
31 \( 1 + (8.12e3 - 1.40e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (1.66e4 + 2.88e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 9.86e4iT - 4.75e9T^{2} \)
43 \( 1 - 6.21e4T + 6.32e9T^{2} \)
47 \( 1 + (-5.03e4 + 2.90e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (1.57e5 + 9.07e4i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-1.27e5 - 7.36e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-6.34e3 - 1.09e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.88e5 - 3.26e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 1.18e5iT - 1.28e11T^{2} \)
73 \( 1 + (-3.41e3 + 5.91e3i)T + (-7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (1.83e5 + 3.18e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 3.36e5iT - 3.26e11T^{2} \)
89 \( 1 + (-2.58e5 + 1.49e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 5.39e5T + 8.32e11T^{2} \)
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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

−17.54276352236775486069389451829, −16.17342254115762899642753201782, −14.58930111022112542058263908137, −13.08025140251501327423289573860, −12.17782757927614272011261875336, −11.05114630476933845526368111177, −8.409074353405843485258873398344, −7.57722577539163715544156647405, −5.38272461774689671253387646548, −2.88793521522403172984709159050, 0.27140740362024891817801246025, 4.26108514131735500626812368977, 5.31463401968775337662784001186, 7.81614166972056075836278137766, 9.788577168061528511969914200554, 10.62372536694518756348270486294, 12.28811820517503317361444154684, 14.07077447637828017224620388235, 15.16616241418053401330187100722, 16.18385793184080086077159532101

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