Properties

Label 2-21-21.5-c3-0-1
Degree $2$
Conductor $21$
Sign $0.790 - 0.612i$
Analytic cond. $1.23904$
Root an. cond. $1.11312$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.65 + 0.954i)2-s + (1.60 + 4.94i)3-s + (−2.17 − 3.77i)4-s + (−0.623 + 1.08i)5-s + (−2.05 + 9.70i)6-s + (10.0 − 15.5i)7-s − 23.5i·8-s + (−21.8 + 15.8i)9-s + (−2.06 + 1.19i)10-s + (−35.2 + 20.3i)11-s + (15.1 − 16.8i)12-s − 19.5i·13-s + (31.4 − 16.0i)14-s + (−6.34 − 1.34i)15-s + (5.08 − 8.80i)16-s + (52.3 + 90.6i)17-s + ⋯
L(s)  = 1  + (0.584 + 0.337i)2-s + (0.309 + 0.950i)3-s + (−0.272 − 0.471i)4-s + (−0.0557 + 0.0966i)5-s + (−0.140 + 0.660i)6-s + (0.544 − 0.838i)7-s − 1.04i·8-s + (−0.808 + 0.588i)9-s + (−0.0652 + 0.0376i)10-s + (−0.965 + 0.557i)11-s + (0.364 − 0.404i)12-s − 0.418i·13-s + (0.601 − 0.306i)14-s + (−0.109 − 0.0231i)15-s + (0.0794 − 0.137i)16-s + (0.746 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.790 - 0.612i$
Analytic conductor: \(1.23904\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3/2),\ 0.790 - 0.612i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.33564 + 0.457034i\)
\(L(\frac12)\) \(\approx\) \(1.33564 + 0.457034i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.60 - 4.94i)T \)
7 \( 1 + (-10.0 + 15.5i)T \)
good2 \( 1 + (-1.65 - 0.954i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (0.623 - 1.08i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (35.2 - 20.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 19.5iT - 2.19e3T^{2} \)
17 \( 1 + (-52.3 - 90.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-35.0 - 20.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (69.6 + 40.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 211. iT - 2.43e4T^{2} \)
31 \( 1 + (86.6 - 50.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-94.9 + 164. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 186.T + 6.89e4T^{2} \)
43 \( 1 - 158.T + 7.95e4T^{2} \)
47 \( 1 + (179. - 310. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-366. + 211. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (312. + 541. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-699. - 403. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (149. + 258. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 455. iT - 3.57e5T^{2} \)
73 \( 1 + (434. - 250. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-30.9 + 53.6i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 73.1T + 5.71e5T^{2} \)
89 \( 1 + (57.3 - 99.3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.41e3iT - 9.12e5T^{2} \)
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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

−17.64313186916079626230625355438, −16.20788436703345701815981337737, −15.04369893279872502174988296771, −14.31187287460262407200707434810, −13.00553313051339671835814174028, −10.74019590217713869346524488833, −9.869899958833306500659034077419, −7.86416069823838860413542642954, −5.49556994590938450967968799317, −4.06378845584135650286032186020, 2.77228712040942353118855861849, 5.35183005861158463724254213436, 7.69231005395256480004243179018, 8.853319795679215224943372075893, 11.49476576352244984147631075603, 12.35980317493511530772783635731, 13.56398794564205116065305681336, 14.48943332057903644605778368371, 16.31928407909573959272038428720, 18.02262054129353343944014106297

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