Properties

Label 2-21-21.11-c6-0-1
Degree $2$
Conductor $21$
Sign $-0.950 + 0.311i$
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  + (−5.69 + 3.28i)2-s + (2.79 + 26.8i)3-s + (−10.3 + 17.9i)4-s + (57.4 − 33.1i)5-s + (−104. − 143. i)6-s + (−329. + 93.7i)7-s − 557. i·8-s + (−713. + 149. i)9-s + (−218. + 377. i)10-s + (−603. − 348. i)11-s + (−511. − 228. i)12-s − 824.·13-s + (1.57e3 − 1.61e3i)14-s + (1.05e3 + 1.45e3i)15-s + (1.16e3 + 2.02e3i)16-s + (4.78e3 + 2.76e3i)17-s + ⋯
L(s)  = 1  + (−0.712 + 0.411i)2-s + (0.103 + 0.994i)3-s + (−0.161 + 0.280i)4-s + (0.459 − 0.265i)5-s + (−0.482 − 0.665i)6-s + (−0.961 + 0.273i)7-s − 1.08i·8-s + (−0.978 + 0.205i)9-s + (−0.218 + 0.377i)10-s + (−0.453 − 0.261i)11-s + (−0.295 − 0.132i)12-s − 0.375·13-s + (0.572 − 0.590i)14-s + (0.311 + 0.429i)15-s + (0.285 + 0.494i)16-s + (0.973 + 0.562i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.950 + 0.311i$
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: \((2,\ 21,\ (\ :3),\ -0.950 + 0.311i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0819491 - 0.512724i\)
\(L(\frac12)\) \(\approx\) \(0.0819491 - 0.512724i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.79 - 26.8i)T \)
7 \( 1 + (329. - 93.7i)T \)
good2 \( 1 + (5.69 - 3.28i)T + (32 - 55.4i)T^{2} \)
5 \( 1 + (-57.4 + 33.1i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (603. + 348. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 824.T + 4.82e6T^{2} \)
17 \( 1 + (-4.78e3 - 2.76e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-2.52e3 - 4.37e3i)T + (-2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (1.72e4 - 9.94e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 2.34e4iT - 5.94e8T^{2} \)
31 \( 1 + (2.31e4 - 4.01e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-2.41e4 - 4.18e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 1.59e4iT - 4.75e9T^{2} \)
43 \( 1 - 2.17e4T + 6.32e9T^{2} \)
47 \( 1 + (-3.30e4 + 1.90e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-1.93e5 - 1.11e5i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (2.71e5 + 1.56e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.26e5 + 2.19e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-8.29e4 + 1.43e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 2.98e5iT - 1.28e11T^{2} \)
73 \( 1 + (2.21e3 - 3.83e3i)T + (-7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-3.10e5 - 5.37e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 4.14e5iT - 3.26e11T^{2} \)
89 \( 1 + (-6.49e5 + 3.75e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 2.12e5T + 8.32e11T^{2} \)
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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

−17.19204913581642049935849915510, −16.39173462900979940845914117352, −15.48580894970293101113400788549, −13.75730545469917561803140526446, −12.29547174300083318023992530114, −10.10548498866010892525274218721, −9.412464850148850191815197405064, −7.986071668307615889651945298428, −5.77611544232410599010687154848, −3.53124341014024103188087685339, 0.39343914491868736277561447384, 2.41224398821020214738173324144, 5.87210979957573175319382304998, 7.55813185169290351184482676314, 9.299386099656732695078415457623, 10.41577438381386390486911882503, 12.11033115298861499275796891026, 13.49849920242713219356101357722, 14.50796183867987130598139643266, 16.53451924941879166792179430566

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