Properties

Label 2-21e2-49.22-c1-0-2
Degree $2$
Conductor $441$
Sign $-0.768 + 0.639i$
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  + (−1.28 + 0.618i)2-s + (0.0219 − 0.0275i)4-s + (−0.813 + 3.56i)5-s + (−0.766 + 2.53i)7-s + (0.623 − 2.73i)8-s + (−1.16 − 5.08i)10-s + (−2.18 + 1.05i)11-s + (−2.46 + 1.18i)13-s + (−0.581 − 3.72i)14-s + (0.905 + 3.96i)16-s + (−0.759 − 0.952i)17-s + 7.04·19-s + (0.0802 + 0.100i)20-s + (2.15 − 2.70i)22-s + (4.18 − 5.25i)23-s + ⋯
L(s)  = 1  + (−0.908 + 0.437i)2-s + (0.0109 − 0.0137i)4-s + (−0.363 + 1.59i)5-s + (−0.289 + 0.957i)7-s + (0.220 − 0.966i)8-s + (−0.367 − 1.60i)10-s + (−0.658 + 0.317i)11-s + (−0.683 + 0.329i)13-s + (−0.155 − 0.996i)14-s + (0.226 + 0.991i)16-s + (−0.184 − 0.231i)17-s + 1.61·19-s + (0.0179 + 0.0225i)20-s + (0.459 − 0.576i)22-s + (0.873 − 1.09i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.768 + 0.639i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.768 + 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126987 - 0.351015i\)
\(L(\frac12)\) \(\approx\) \(0.126987 - 0.351015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 - 2.53i)T \)
good2 \( 1 + (1.28 - 0.618i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (0.813 - 3.56i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (2.18 - 1.05i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.46 - 1.18i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (0.759 + 0.952i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 - 7.04T + 19T^{2} \)
23 \( 1 + (-4.18 + 5.25i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-0.0106 - 0.0133i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + 9.28T + 31T^{2} \)
37 \( 1 + (2.51 + 3.15i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (0.696 - 3.04i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-0.183 - 0.803i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (8.73 - 4.20i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-3.39 + 4.25i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-1.00 - 4.38i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (0.616 + 0.773i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 - 4.50T + 67T^{2} \)
71 \( 1 + (8.59 - 10.7i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (3.23 + 1.55i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 9.66T + 79T^{2} \)
83 \( 1 + (-8.40 - 4.04i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-14.5 - 7.01i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 - 8.30T + 97T^{2} \)
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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

−11.48474417153653492402825410624, −10.57184852606632411826975514580, −9.731964162724115564082520230362, −9.042657147361305476995108325705, −7.84797565903315330329928188385, −7.20507144918356722517957238742, −6.52875429715874889029062999367, −5.13837973056991659017133597775, −3.48280765516366076528482693722, −2.53259360799702187590556275711, 0.33122811958730155241440422773, 1.46396342058785747036185309951, 3.43983483948769288051518155099, 4.92895588317371658009656752281, 5.39751762114251533341230076254, 7.39041433994012578899583406697, 7.939910405166534010572183514312, 8.998237108897547531461146657864, 9.542932675687122907305076753845, 10.41326305579528994680938661039

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