Properties

Label 2-21e2-21.5-c3-0-20
Degree $2$
Conductor $441$
Sign $0.778 - 0.627i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
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  + (1.57 + 0.910i)2-s + (−2.34 − 4.05i)4-s + (−7.54 + 13.0i)5-s − 23.0i·8-s + (−23.7 + 13.7i)10-s + (8.56 − 4.94i)11-s − 67.8i·13-s + (2.27 − 3.93i)16-s + (35.0 + 60.7i)17-s + (53.2 + 30.7i)19-s + 70.7·20-s + 18.0·22-s + (113. + 65.7i)23-s + (−51.3 − 88.8i)25-s + (61.7 − 107. i)26-s + ⋯
L(s)  = 1  + (0.557 + 0.321i)2-s + (−0.292 − 0.507i)4-s + (−0.674 + 1.16i)5-s − 1.02i·8-s + (−0.752 + 0.434i)10-s + (0.234 − 0.135i)11-s − 1.44i·13-s + (0.0355 − 0.0615i)16-s + (0.500 + 0.866i)17-s + (0.642 + 0.371i)19-s + 0.790·20-s + 0.174·22-s + (1.03 + 0.596i)23-s + (−0.410 − 0.711i)25-s + (0.466 − 0.807i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.778 - 0.627i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.778 - 0.627i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.059556319\)
\(L(\frac12)\) \(\approx\) \(2.059556319\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-1.57 - 0.910i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (7.54 - 13.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-8.56 + 4.94i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 67.8iT - 2.19e3T^{2} \)
17 \( 1 + (-35.0 - 60.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-53.2 - 30.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-113. - 65.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 158. iT - 2.43e4T^{2} \)
31 \( 1 + (-66.2 + 38.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (174. - 301. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 138.T + 6.89e4T^{2} \)
43 \( 1 - 539.T + 7.95e4T^{2} \)
47 \( 1 + (-111. + 193. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-459. + 265. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-271. - 470. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-116. - 67.0i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (160. + 277. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 416. iT - 3.57e5T^{2} \)
73 \( 1 + (472. - 272. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-161. + 279. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 885.T + 5.71e5T^{2} \)
89 \( 1 + (-812. + 1.40e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 739. iT - 9.12e5T^{2} \)
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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

−10.54972398323479657506806974242, −10.29625153520183658839687164095, −8.989345118920122945023496833506, −7.77462902914007745861279506216, −7.03080285175614448362427431516, −5.99279891071463242542492158945, −5.19404885417423976064274342232, −3.78310600917470060203299546731, −3.09139190287186938573367495421, −0.974010489005056213576194949680, 0.77958743585801421946128310638, 2.51792118825530157534471730859, 3.95405226611735862906662311006, 4.53806058012482604439219107525, 5.43096131548632689739620788066, 7.04810378547477394532240694890, 7.901961282432339602395946395450, 9.008475249509043586330934993072, 9.275042404508881168543669655596, 10.98917144601985574552140737453

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