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Properties

Label 2-21e2-63.47-c1-0-17
Degree $2$
Conductor $441$
Sign $0.110 - 0.993i$
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  + (−1.5 + 0.866i)2-s + (1.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + 3·5-s − 3·6-s − 1.73i·8-s + (1.5 + 2.59i)9-s + (−4.5 + 2.59i)10-s + 1.73i·11-s + (1.5 − 0.866i)12-s + (1.5 − 0.866i)13-s + (4.5 + 2.59i)15-s + (2.49 + 4.33i)16-s + (−1.5 − 2.59i)17-s + (−4.5 − 2.59i)18-s + (4.5 + 2.59i)19-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + (0.866 + 0.499i)3-s + (0.250 − 0.433i)4-s + 1.34·5-s − 1.22·6-s − 0.612i·8-s + (0.5 + 0.866i)9-s + (−1.42 + 0.821i)10-s + 0.522i·11-s + (0.433 − 0.250i)12-s + (0.416 − 0.240i)13-s + (1.16 + 0.670i)15-s + (0.624 + 1.08i)16-s + (−0.363 − 0.630i)17-s + (−1.06 − 0.612i)18-s + (1.03 + 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.110 - 0.993i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.110 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995346 + 0.890429i\)
\(L(\frac12)\) \(\approx\) \(0.995346 + 0.890429i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 + (4.5 + 2.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 - 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.5 - 4.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12 - 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (-4.5 + 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.5 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.70704290299500632395616395743, −9.966812068373971656602248906909, −9.489066931635769291659851217351, −8.758203295864613541670595723170, −7.87622207116831882661162770878, −6.95026303310152955015798327225, −5.88170551328094744574311991043, −4.60296041835022587449067178750, −3.10915928561486700508227992890, −1.68506314583233985598526436150, 1.31476918920905115884516405007, 2.18008246770824568750489151264, 3.38975232692628683184736471929, 5.33625661556151440208566839580, 6.34762994543872814409830464126, 7.55199664669973416981709926685, 8.508188353993272298851908614266, 9.339725434933373327355123457551, 9.622970219408949043899417321996, 10.71430142050717045548279579408

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