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Properties

Label 2-21-21.11-c4-0-5
Degree $2$
Conductor $21$
Sign $0.988 - 0.149i$
Analytic cond. $2.17076$
Root an. cond. $1.47335$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 − 0.776i)2-s + (8.38 + 3.27i)3-s + (−6.79 + 11.7i)4-s + (23.4 − 13.5i)5-s + (13.8 − 2.10i)6-s + (−27.0 − 40.8i)7-s + 45.9i·8-s + (59.5 + 54.8i)9-s + (20.9 − 36.3i)10-s + (−130. − 75.1i)11-s + (−95.4 + 76.4i)12-s − 189.·13-s + (−68.1 − 33.8i)14-s + (240. − 36.6i)15-s + (−73.0 − 126. i)16-s + (136. + 79.0i)17-s + ⋯
L(s)  = 1  + (0.336 − 0.194i)2-s + (0.931 + 0.363i)3-s + (−0.424 + 0.735i)4-s + (0.936 − 0.540i)5-s + (0.383 − 0.0585i)6-s + (−0.552 − 0.833i)7-s + 0.717i·8-s + (0.735 + 0.677i)9-s + (0.209 − 0.363i)10-s + (−1.07 − 0.621i)11-s + (−0.663 + 0.530i)12-s − 1.12·13-s + (−0.347 − 0.172i)14-s + (1.06 − 0.163i)15-s + (−0.285 − 0.494i)16-s + (0.473 + 0.273i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.988 - 0.149i$
Analytic conductor: \(2.17076\)
Root analytic conductor: \(1.47335\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :2),\ 0.988 - 0.149i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.78828 + 0.134371i\)
\(L(\frac12)\) \(\approx\) \(1.78828 + 0.134371i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.38 - 3.27i)T \)
7 \( 1 + (27.0 + 40.8i)T \)
good2 \( 1 + (-1.34 + 0.776i)T + (8 - 13.8i)T^{2} \)
5 \( 1 + (-23.4 + 13.5i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (130. + 75.1i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 189.T + 2.85e4T^{2} \)
17 \( 1 + (-136. - 79.0i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-67.5 - 116. i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-791. + 456. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 177. iT - 7.07e5T^{2} \)
31 \( 1 + (134. - 233. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-205. - 356. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 1.72e3iT - 2.82e6T^{2} \)
43 \( 1 - 811.T + 3.41e6T^{2} \)
47 \( 1 + (211. - 122. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-1.86e3 - 1.07e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (1.22e3 + 705. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (938. + 1.62e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.00e3 + 1.73e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 5.33e3iT - 2.54e7T^{2} \)
73 \( 1 + (2.60e3 - 4.50e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (4.93e3 + 8.54e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 9.01e3iT - 4.74e7T^{2} \)
89 \( 1 + (-1.13e4 + 6.54e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 7.27e3T + 8.85e7T^{2} \)
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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

−17.16426090265349313492963409738, −16.36720896678180477413834617474, −14.48728697012175434392650936335, −13.38357665171614869217738168246, −12.80377749102902308897344396535, −10.36169846137816387682114422881, −9.133177488261754561880130009060, −7.68970410564976882908157092862, −4.89243400401016631768142934658, −2.99545049964791453773188517067, 2.52784668260452574388921526710, 5.41154980973924303426802446441, 7.09826879882450614527396165326, 9.303517666054644407898856086249, 10.06251104728144803347000308829, 12.66895035068621976246949555434, 13.60776965092880445618228041804, 14.75110918670155177326500042657, 15.49736130968222059624902078380, 17.78446981898028967964493082450

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