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Properties

Label 2-21e2-49.29-c1-0-6
Degree $2$
Conductor $441$
Sign $0.342 + 0.939i$
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  + (−2.24 − 1.08i)2-s + (2.61 + 3.28i)4-s + (0.222 + 0.974i)5-s + (−2.64 + 0.0800i)7-s + (−1.21 − 5.33i)8-s + (0.553 − 2.42i)10-s + (−1.23 − 0.594i)11-s + (−3.37 − 1.62i)13-s + (6.01 + 2.67i)14-s + (−1.16 + 5.08i)16-s + (2.96 − 3.72i)17-s + 4.98·19-s + (−2.61 + 3.27i)20-s + (2.12 + 2.66i)22-s + (3.73 + 4.67i)23-s + ⋯
L(s)  = 1  + (−1.58 − 0.763i)2-s + (1.30 + 1.64i)4-s + (0.0994 + 0.435i)5-s + (−0.999 + 0.0302i)7-s + (−0.430 − 1.88i)8-s + (0.175 − 0.766i)10-s + (−0.372 − 0.179i)11-s + (−0.937 − 0.451i)13-s + (1.60 + 0.715i)14-s + (−0.290 + 1.27i)16-s + (0.719 − 0.902i)17-s + 1.14·19-s + (−0.584 + 0.733i)20-s + (0.453 + 0.568i)22-s + (0.778 + 0.975i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438978 - 0.307377i\)
\(L(\frac12)\) \(\approx\) \(0.438978 - 0.307377i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0800i)T \)
good2 \( 1 + (2.24 + 1.08i)T + (1.24 + 1.56i)T^{2} \)
5 \( 1 + (-0.222 - 0.974i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (1.23 + 0.594i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (3.37 + 1.62i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-2.96 + 3.72i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 - 4.98T + 19T^{2} \)
23 \( 1 + (-3.73 - 4.67i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-4.13 + 5.18i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 - 8.43T + 31T^{2} \)
37 \( 1 + (3.36 - 4.21i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (0.231 + 1.01i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-2.13 + 9.35i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-3.97 - 1.91i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (8.27 + 10.3i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-1.11 + 4.88i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (0.916 - 1.14i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 - 7.39T + 67T^{2} \)
71 \( 1 + (-4.50 - 5.64i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (13.6 - 6.58i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + 4.76T + 79T^{2} \)
83 \( 1 + (-11.5 + 5.54i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (8.28 - 3.98i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 - 7.04T + 97T^{2} \)
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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

−10.60615254170545521000450929189, −9.865040277815322007616865319677, −9.568680810822954734150415832898, −8.377092662513409637564044652108, −7.45405590814878317276833754709, −6.78054894773126033740738675276, −5.27519766600270134124901814369, −3.18589318192031073042806915305, −2.67798153386980501702260219538, −0.71357024244322407883840101098, 1.07216098437741133543908940110, 2.87099336304884367230623564614, 4.86261151531722500353162106460, 6.04992932399699606706162559840, 6.91205049539836531833113802585, 7.67303635347786870155573150000, 8.690749529472264229225646848806, 9.385873815988263100822894390158, 10.09051584476483577188422481737, 10.74926340531753543602870389758

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