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Properties

Label 2-21e2-7.3-c2-0-17
Degree $2$
Conductor $441$
Sign $0.0633 - 0.997i$
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 + 2.59i)2-s + (−2.5 + 4.33i)4-s + (4.5 − 2.59i)5-s − 3.00·8-s + (13.5 + 7.79i)10-s + (7.5 − 12.9i)11-s + 13.8i·13-s + (5.49 + 9.52i)16-s + (9 + 5.19i)17-s + (9 − 5.19i)19-s + 25.9i·20-s + 45·22-s + (1 − 1.73i)25-s + (−36 + 20.7i)26-s + 9·29-s + ⋯
L(s)  = 1  + (0.750 + 1.29i)2-s + (−0.625 + 1.08i)4-s + (0.900 − 0.519i)5-s − 0.375·8-s + (1.35 + 0.779i)10-s + (0.681 − 1.18i)11-s + 1.06i·13-s + (0.343 + 0.595i)16-s + (0.529 + 0.305i)17-s + (0.473 − 0.273i)19-s + 1.29i·20-s + 2.04·22-s + (0.0400 − 0.0692i)25-s + (−1.38 + 0.799i)26-s + 0.310·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.0633 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.31850 + 2.17603i\)
\(L(\frac12)\) \(\approx\) \(2.31850 + 2.17603i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-1.5 - 2.59i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + (-4.5 + 2.59i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-7.5 + 12.9i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 13.8iT - 169T^{2} \)
17 \( 1 + (-9 - 5.19i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-9 + 5.19i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 9T + 841T^{2} \)
31 \( 1 + (-10.5 - 6.06i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 10.3iT - 1.68e3T^{2} \)
43 \( 1 + 74T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-16.5 + 28.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-13.5 - 7.79i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (78 - 45.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38 + 65.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 84T + 5.04e3T^{2} \)
73 \( 1 + (-54 - 31.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-21.5 - 37.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 119. iT - 6.88e3T^{2} \)
89 \( 1 + (63 - 36.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 185. iT - 9.40e3T^{2} \)
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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

−11.28239680886991323352204439331, −10.02819274721080405891246255996, −9.021821335528425036097809434405, −8.339846893183846490102728546834, −7.10813269465049303391730710493, −6.26736350532460625388042975350, −5.60668829887881067999572827009, −4.66743094418123392648240914773, −3.49244339944580300799412231899, −1.49820219931075731157811978334, 1.36584315168770142526077356479, 2.48541888439104750421038234984, 3.45177764469321416326660028116, 4.69408125065792332282838004798, 5.62549660869494719576336807405, 6.78706665317338901335409424432, 7.925207923471441567100627133405, 9.525546350238585802318926224962, 10.02089355902050954538393607182, 10.65184406160002211584628549957

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