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Properties

Label 2-21-21.11-c22-0-19
Degree $2$
Conductor $21$
Sign $0.0573 - 0.998i$
Analytic cond. $64.4085$
Root an. cond. $8.02549$
Motivic weight $22$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.85e4 − 1.53e5i)3-s + (−2.09e6 + 3.63e6i)4-s + (9.98e8 + 1.70e9i)7-s + (−1.56e10 − 2.71e10i)9-s + (3.71e11 + 6.43e11i)12-s + 2.09e12·13-s + (−8.79e12 − 1.52e13i)16-s + (−1.03e13 − 1.79e13i)19-s + (3.50e14 − 2.11e12i)21-s + (−1.19e15 + 2.06e15i)25-s − 5.55e15·27-s + (−8.29e15 + 4.99e13i)28-s + (−1.85e16 + 3.21e16i)31-s + (1.31e17 + 8i)36-s + (−4.22e16 − 7.31e16i)37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.505 + 0.862i)7-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)12-s + 1.16·13-s + (−0.499 − 0.866i)16-s + (−0.0887 − 0.153i)19-s + (0.999 − 0.00602i)21-s + (−0.499 + 0.866i)25-s − 27-s + (−0.999 + 0.00602i)28-s + (−0.729 + 1.26i)31-s + 36-s + (−0.237 − 0.411i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0573 - 0.998i)\, \overline{\Lambda}(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & (0.0573 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.0573 - 0.998i$
Analytic conductor: \(64.4085\)
Root analytic conductor: \(8.02549\)
Motivic weight: \(22\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :11),\ 0.0573 - 0.998i)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(1.745498070\)
\(L(\frac12)\) \(\approx\) \(1.745498070\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.85e4 + 1.53e5i)T \)
7 \( 1 + (-9.98e8 - 1.70e9i)T \)
good2 \( 1 + (2.09e6 - 3.63e6i)T^{2} \)
5 \( 1 + (1.19e15 - 2.06e15i)T^{2} \)
11 \( 1 + (4.07e22 + 7.04e22i)T^{2} \)
13 \( 1 - 2.09e12T + 3.21e24T^{2} \)
17 \( 1 + (5.87e26 + 1.01e27i)T^{2} \)
19 \( 1 + (1.03e13 + 1.79e13i)T + (-6.78e27 + 1.17e28i)T^{2} \)
23 \( 1 + (4.53e29 - 7.86e29i)T^{2} \)
29 \( 1 - 1.48e32T^{2} \)
31 \( 1 + (1.85e16 - 3.21e16i)T + (-3.22e32 - 5.59e32i)T^{2} \)
37 \( 1 + (4.22e16 + 7.31e16i)T + (-1.58e34 + 2.74e34i)T^{2} \)
41 \( 1 - 3.02e35T^{2} \)
43 \( 1 - 1.26e18T + 8.63e35T^{2} \)
47 \( 1 + (3.05e36 - 5.29e36i)T^{2} \)
53 \( 1 + (4.29e37 + 7.44e37i)T^{2} \)
59 \( 1 + (4.54e38 + 7.87e38i)T^{2} \)
61 \( 1 + (-3.36e19 - 5.83e19i)T + (-9.46e38 + 1.63e39i)T^{2} \)
67 \( 1 + (1.07e20 - 1.85e20i)T + (-7.45e39 - 1.29e40i)T^{2} \)
71 \( 1 - 5.34e40T^{2} \)
73 \( 1 + (3.10e20 - 5.38e20i)T + (-4.92e40 - 8.52e40i)T^{2} \)
79 \( 1 + (-5.18e20 - 8.98e20i)T + (-2.79e41 + 4.84e41i)T^{2} \)
83 \( 1 - 1.65e42T^{2} \)
89 \( 1 + (3.85e42 - 6.66e42i)T^{2} \)
97 \( 1 + 1.61e21T + 5.11e43T^{2} \)
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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

−13.37517800005923698767915862728, −12.44653999021696057947110473897, −11.36621168039621992412375261579, −9.065689293554382311384110203527, −8.411835390081561262960752756214, −7.22247068488377560410173538599, −5.62837798200384204957780300273, −3.82347155259651740531913589461, −2.62096946047486632472174030516, −1.27399615008367403180032279431, 0.42523232211675483034127245658, 1.81991798293470538725445347445, 3.73641341379256289259871733339, 4.63385109819513583727593550331, 5.99321389419499714149232001011, 7.971420913561324854088479446579, 9.172345541004213088969310451312, 10.32887734908971639787694117015, 11.12816489753663890751407393301, 13.43683224837033743629287262401

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