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Properties

Label 2-21e2-9.5-c2-0-48
Degree $2$
Conductor $441$
Sign $0.642 + 0.766i$
Analytic cond. $12.0163$
Root an. cond. $3.46646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)2-s + (1.5 + 2.59i)3-s + (−0.5 + 0.866i)4-s + (−3 − 1.73i)5-s + (−4.5 − 2.59i)6-s − 8.66i·8-s + (−4.5 + 7.79i)9-s + 6·10-s + (−1.5 + 0.866i)11-s − 2.99·12-s + (−2 + 3.46i)13-s − 10.3i·15-s + (5.5 + 9.52i)16-s − 15.5i·17-s − 15.5i·18-s − 11·19-s + ⋯
L(s)  = 1  + (−0.750 + 0.433i)2-s + (0.5 + 0.866i)3-s + (−0.125 + 0.216i)4-s + (−0.600 − 0.346i)5-s + (−0.750 − 0.433i)6-s − 1.08i·8-s + (−0.5 + 0.866i)9-s + 0.600·10-s + (−0.136 + 0.0787i)11-s − 0.249·12-s + (−0.153 + 0.266i)13-s − 0.692i·15-s + (0.343 + 0.595i)16-s − 0.916i·17-s − 0.866i·18-s − 0.578·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.368668 - 0.171913i\)
\(L(\frac12)\) \(\approx\) \(0.368668 - 0.171913i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 2.59i)T \)
7 \( 1 \)
good2 \( 1 + (1.5 - 0.866i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (3 + 1.73i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 15.5iT - 289T^{2} \)
19 \( 1 + 11T + 361T^{2} \)
23 \( 1 + (24 + 13.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-39 + 22.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-16 + 27.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 34T + 1.36e3T^{2} \)
41 \( 1 + (-10.5 - 6.06i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-30.5 - 52.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-42 + 24.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (43.5 + 25.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-28 - 48.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-15.5 + 26.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 31.1iT - 5.04e3T^{2} \)
73 \( 1 + 65T + 5.32e3T^{2} \)
79 \( 1 + (19 + 32.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-42 + 24.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 124. iT - 7.92e3T^{2} \)
97 \( 1 + (57.5 + 99.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.37088454649139998465147996706, −9.745409316788149028732132121849, −8.847846943763536574564492713897, −8.191906145016284461504504462791, −7.53208387473437966742227349165, −6.23376129906621284834969570181, −4.61153630673139266162736789340, −4.06655112045638403606975738569, −2.65926086902810485770814681841, −0.22025839297479836410119128447, 1.31628506109092044566817564008, 2.54086999980883624873988362677, 3.83239087844297726504003168005, 5.46204472324425008586287315441, 6.57457084138698420347722117444, 7.66887425646739885631631514050, 8.357742853069927303723483973323, 9.060951390931452355521991858415, 10.22715415544686596910180834666, 10.85365429903368899682285319279

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