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Properties

Label 2-21e2-63.13-c2-0-18
Degree $2$
Conductor $441$
Sign $-0.307 + 0.951i$
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  + (−1.67 − 2.90i)2-s + (−1.29 − 2.70i)3-s + (−3.62 + 6.28i)4-s + (0.769 + 0.444i)5-s + (−5.68 + 8.30i)6-s + 10.9·8-s + (−5.64 + 7.01i)9-s − 2.98i·10-s + (2.75 + 4.76i)11-s + (21.7 + 1.67i)12-s + (12.3 + 7.11i)13-s + (0.205 − 2.65i)15-s + (−3.81 − 6.61i)16-s − 13.0i·17-s + (29.8 + 4.63i)18-s + 26.2i·19-s + ⋯
L(s)  = 1  + (−0.838 − 1.45i)2-s + (−0.431 − 0.901i)3-s + (−0.907 + 1.57i)4-s + (0.153 + 0.0888i)5-s + (−0.948 + 1.38i)6-s + 1.36·8-s + (−0.627 + 0.778i)9-s − 0.298i·10-s + (0.250 + 0.433i)11-s + (1.80 + 0.139i)12-s + (0.948 + 0.547i)13-s + (0.0136 − 0.177i)15-s + (−0.238 − 0.413i)16-s − 0.766i·17-s + (1.65 + 0.257i)18-s + 1.38i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.307 + 0.951i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ -0.307 + 0.951i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.517605 - 0.711620i\)
\(L(\frac12)\) \(\approx\) \(0.517605 - 0.711620i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.29 + 2.70i)T \)
7 \( 1 \)
good2 \( 1 + (1.67 + 2.90i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + (-0.769 - 0.444i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-2.75 - 4.76i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-12.3 - 7.11i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 13.0iT - 289T^{2} \)
19 \( 1 - 26.2iT - 361T^{2} \)
23 \( 1 + (-18.0 + 31.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-16.4 - 28.5i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-32.8 - 18.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 42.9T + 1.36e3T^{2} \)
41 \( 1 + (-51.1 - 29.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (14.4 + 25.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (10.8 - 6.27i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 9.76T + 2.80e3T^{2} \)
59 \( 1 + (-21.3 - 12.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.7 - 9.08i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.1 - 26.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 43.4T + 5.04e3T^{2} \)
73 \( 1 - 27.2iT - 5.32e3T^{2} \)
79 \( 1 + (-33.7 - 58.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-8.30 + 4.79i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 63.3iT - 7.92e3T^{2} \)
97 \( 1 + (-76.9 + 44.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.65668463896048501722791868425, −10.05858397500038476488942048494, −8.822243557182461173378754993429, −8.292155925734176719826273689342, −7.03979136139126957451473788507, −6.09370202275472909873008212802, −4.52593237766330329997523667541, −3.08015848259058070467435322153, −1.93065217344468899028971014495, −0.915266079448992479371087030423, 0.76318078386830230956626131047, 3.43983448861708490691627397744, 4.83882997408878378546122931973, 5.79335704621493938279388935816, 6.35566205503225673622992074126, 7.54849025996628202969313696442, 8.592666438476820622893886533344, 9.153403505922036773634083536078, 9.975715751974002676439307581980, 10.90723929498857349660234184131

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