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Properties

Label 2-21e2-49.36-c1-0-21
Degree $2$
Conductor $441$
Sign $-0.880 - 0.473i$
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  + (−0.368 − 1.61i)2-s + (−0.671 + 0.323i)4-s + (−0.768 − 0.963i)5-s + (−1.85 − 1.88i)7-s + (−1.29 − 1.62i)8-s + (−1.27 + 1.59i)10-s + (0.480 + 2.10i)11-s + (−0.494 − 2.16i)13-s + (−2.35 + 3.69i)14-s + (−3.07 + 3.85i)16-s + (1.21 + 0.583i)17-s − 1.74·19-s + (0.826 + 0.398i)20-s + (3.22 − 1.55i)22-s + (−3.01 + 1.45i)23-s + ⋯
L(s)  = 1  + (−0.260 − 1.14i)2-s + (−0.335 + 0.161i)4-s + (−0.343 − 0.430i)5-s + (−0.702 − 0.711i)7-s + (−0.458 − 0.574i)8-s + (−0.402 + 0.504i)10-s + (0.144 + 0.635i)11-s + (−0.137 − 0.601i)13-s + (−0.629 + 0.987i)14-s + (−0.769 + 0.964i)16-s + (0.293 + 0.141i)17-s − 0.399·19-s + (0.184 + 0.0890i)20-s + (0.687 − 0.331i)22-s + (−0.628 + 0.302i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.880 - 0.473i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.880 - 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175771 + 0.697627i\)
\(L(\frac12)\) \(\approx\) \(0.175771 + 0.697627i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.85 + 1.88i)T \)
good2 \( 1 + (0.368 + 1.61i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (0.768 + 0.963i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (-0.480 - 2.10i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.494 + 2.16i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-1.21 - 0.583i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 + 1.74T + 19T^{2} \)
23 \( 1 + (3.01 - 1.45i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (6.38 + 3.07i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 6.80T + 31T^{2} \)
37 \( 1 + (-5.84 - 2.81i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (0.915 + 1.14i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-5.27 + 6.61i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.595 - 2.60i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-12.0 + 5.81i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-3.23 + 4.05i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (14.0 + 6.74i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 - 4.18T + 67T^{2} \)
71 \( 1 + (-13.7 + 6.61i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (0.565 - 2.47i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 - 0.184T + 79T^{2} \)
83 \( 1 + (0.579 - 2.53i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-3.12 + 13.6i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−10.57144663585218879570535409523, −9.879577651249449184403926022944, −9.191936439307831613620632885812, −7.942235012225289582235872717533, −6.97231784629229639917004620988, −5.83330963478110646334998376062, −4.26451282867748539258638728428, −3.47530960389435908024147064477, −2.06811855849162343600460029381, −0.46677730851040940829147961712, 2.51603591737333487977508258925, 3.76699386520717535873623504922, 5.43975493796399842857154089660, 6.14842703483226816779930559948, 7.03266251587314105781725464888, 7.79730088405791841052307908969, 8.904274219503417283663149155315, 9.385105575249975856797675982697, 10.81792301589274722525252095954, 11.59520597514272965762725846021

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