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Properties

Label 2-21e2-21.17-c1-0-4
Degree $2$
Conductor $441$
Sign $0.932 - 0.360i$
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  + (−2.23 + 1.28i)2-s + (2.32 − 4.02i)4-s + 6.82i·8-s + (−0.790 − 0.456i)11-s + (−4.14 − 7.18i)16-s + 2.35·22-s + (8.13 − 4.69i)23-s + (2.5 − 4.33i)25-s − 6.06i·29-s + (6.69 + 3.86i)32-s + (5.29 + 9.16i)37-s + 5.29·43-s + (−3.67 + 2.12i)44-s + (−12.1 + 20.9i)46-s + 12.8i·50-s + ⋯
L(s)  = 1  + (−1.57 + 0.911i)2-s + (1.16 − 2.01i)4-s + 2.41i·8-s + (−0.238 − 0.137i)11-s + (−1.03 − 1.79i)16-s + 0.501·22-s + (1.69 − 0.979i)23-s + (0.5 − 0.866i)25-s − 1.12i·29-s + (1.18 + 0.683i)32-s + (0.869 + 1.50i)37-s + 0.806·43-s + (−0.553 + 0.319i)44-s + (−1.78 + 3.09i)46-s + 1.82i·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.932 - 0.360i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.932 - 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.622121 + 0.115890i\)
\(L(\frac12)\) \(\approx\) \(0.622121 + 0.115890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (2.23 - 1.28i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.790 + 0.456i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-8.13 + 4.69i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.06iT - 29T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.29 - 9.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-12.6 - 7.27i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.57iT - 71T^{2} \)
73 \( 1 + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.74127547343177895110831803396, −10.12448323966940869526072377837, −9.149097884760221563157324182795, −8.482458327524167880791473053859, −7.64343625290946955162861448301, −6.73715615188778955524220298303, −5.95591732567019373887074852064, −4.69979283645166430609544674704, −2.59253670525059862788338845986, −0.843942813266897962547258926178, 1.12837556752328189142159800884, 2.53785582265413435123503991594, 3.63316581285470163299655635133, 5.32950282586308438155516165758, 7.00872286366117936992351920562, 7.54679954755330860192303444988, 8.745298583465036824918541758524, 9.222126309956977554142075979738, 10.15539769895172519161380998300, 11.00842289534447055791734501116

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