Properties

Label 2-21e2-63.5-c1-0-11
Degree $2$
Conductor $441$
Sign $0.972 - 0.233i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.664i·2-s + (−1.15 − 1.28i)3-s + 1.55·4-s + (−0.0141 − 0.0245i)5-s + (0.855 − 0.768i)6-s + 2.36i·8-s + (−0.320 + 2.98i)9-s + (0.0162 − 0.00940i)10-s + (0.885 + 0.511i)11-s + (−1.80 − 2.00i)12-s + (4.87 + 2.81i)13-s + (−0.0152 + 0.0466i)15-s + 1.54·16-s + (−2.83 − 4.91i)17-s + (−1.98 − 0.212i)18-s + (1.81 + 1.04i)19-s + ⋯
L(s)  = 1  + 0.469i·2-s + (−0.668 − 0.743i)3-s + 0.779·4-s + (−0.00632 − 0.0109i)5-s + (0.349 − 0.313i)6-s + 0.835i·8-s + (−0.106 + 0.994i)9-s + (0.00514 − 0.00297i)10-s + (0.266 + 0.154i)11-s + (−0.520 − 0.579i)12-s + (1.35 + 0.781i)13-s + (−0.00392 + 0.0120i)15-s + 0.386·16-s + (−0.688 − 1.19i)17-s + (−0.467 − 0.0501i)18-s + (0.415 + 0.240i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.972 - 0.233i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.972 - 0.233i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44611 + 0.171385i\)
\(L(\frac12)\) \(\approx\) \(1.44611 + 0.171385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.15 + 1.28i)T \)
7 \( 1 \)
good2 \( 1 - 0.664iT - 2T^{2} \)
5 \( 1 + (0.0141 + 0.0245i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.885 - 0.511i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.87 - 2.81i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.83 + 4.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.81 - 1.04i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.28 + 3.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.52 - 2.03i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.31iT - 31T^{2} \)
37 \( 1 + (-1.23 + 2.14i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.52 - 6.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.15 + 2.00i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + (10.0 - 5.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.02T + 59T^{2} \)
61 \( 1 + 2.36iT - 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 7.93iT - 71T^{2} \)
73 \( 1 + (9.43 - 5.44i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + (3.07 + 5.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.02 - 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.77 - 3.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.17321714571469966009235595213, −10.70803112288183538453302853699, −9.134614567648251799143737363977, −8.232357865581792531149112902122, −7.05095471793866978251673768557, −6.70342637536226027437822368105, −5.74745804083339650245348383741, −4.64276867764105366179361033761, −2.79378825979430452801756090500, −1.39414292433799211927294722385, 1.27464977236677586658540948724, 3.17303589641135173305423957047, 3.96125907541730315714048793480, 5.48462775689656327180141767098, 6.22418613558887908042715573548, 7.20793567640169999647135636281, 8.584510043960259715818164387834, 9.509060099933643628442374530409, 10.54566497781367322559407406746, 11.09007933598260148762025466874

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