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Properties

Label 2-21e2-21.5-c3-0-16
Degree $2$
Conductor $441$
Sign $0.901 + 0.432i$
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  + (−0.555 − 0.320i)2-s + (−3.79 − 6.57i)4-s + (−6.03 + 10.4i)5-s + 9.99i·8-s + (6.70 − 3.86i)10-s + (−38.0 + 21.9i)11-s − 66.9i·13-s + (−27.1 + 47.0i)16-s + (−19.7 − 34.1i)17-s + (50.1 + 28.9i)19-s + 91.6·20-s + 28.1·22-s + (39.9 + 23.0i)23-s + (−10.3 − 17.9i)25-s + (−21.4 + 37.1i)26-s + ⋯
L(s)  = 1  + (−0.196 − 0.113i)2-s + (−0.474 − 0.821i)4-s + (−0.539 + 0.935i)5-s + 0.441i·8-s + (0.211 − 0.122i)10-s + (−1.04 + 0.602i)11-s − 1.42i·13-s + (−0.424 + 0.734i)16-s + (−0.281 − 0.487i)17-s + (0.605 + 0.349i)19-s + 1.02·20-s + 0.272·22-s + (0.362 + 0.209i)23-s + (−0.0830 − 0.143i)25-s + (−0.161 + 0.280i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.901 + 0.432i$
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: \((2,\ 441,\ (\ :3/2),\ 0.901 + 0.432i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.031607048\)
\(L(\frac12)\) \(\approx\) \(1.031607048\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.555 + 0.320i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (6.03 - 10.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (38.0 - 21.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 66.9iT - 2.19e3T^{2} \)
17 \( 1 + (19.7 + 34.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-50.1 - 28.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-39.9 - 23.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 201. iT - 2.43e4T^{2} \)
31 \( 1 + (160. - 92.7i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-205. + 355. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 408.T + 6.89e4T^{2} \)
43 \( 1 - 129.T + 7.95e4T^{2} \)
47 \( 1 + (-257. + 445. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-378. + 218. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (303. + 526. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (21.2 + 12.2i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (220. + 382. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.15e3iT - 3.57e5T^{2} \)
73 \( 1 + (-454. + 262. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-75.9 + 131. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.30e3T + 5.71e5T^{2} \)
89 \( 1 + (291. - 505. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.09e3iT - 9.12e5T^{2} \)
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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

−10.77114140835654120230341030384, −9.904275866433008552936190715227, −8.960206190496263157346998977311, −7.74176124412413974319144082697, −7.15455462506882080607252676453, −5.69680849885257260406752195313, −5.04433562111744936900499559321, −3.55739391027398193065755230043, −2.39285465037701001066562751932, −0.60729275627325946525763895852, 0.71202980208608670507668559970, 2.66831852628907746404801225883, 4.10815051984501219840340910860, 4.67187624663126023518890503314, 6.04906590035453455024349126875, 7.43441755289064884692540318232, 8.050764311881591476696283027997, 8.927882100587738632085262966309, 9.481602035052296216449291783501, 10.91144029821073172445734080277

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