Properties

Label 2-21e2-49.8-c1-0-1
Degree $2$
Conductor $441$
Sign $0.800 - 0.599i$
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  + (−1.18 − 1.48i)2-s + (−0.358 + 1.56i)4-s + (0.295 − 0.142i)5-s + (−2.01 + 1.71i)7-s + (−0.667 + 0.321i)8-s + (−0.561 − 0.270i)10-s + (0.320 + 0.402i)11-s + (−0.333 − 0.417i)13-s + (4.93 + 0.955i)14-s + (4.16 + 2.00i)16-s + (1.75 + 7.70i)17-s − 5.08·19-s + (0.117 + 0.514i)20-s + (0.217 − 0.952i)22-s + (0.0438 − 0.192i)23-s + ⋯
L(s)  = 1  + (−0.837 − 1.05i)2-s + (−0.179 + 0.784i)4-s + (0.132 − 0.0636i)5-s + (−0.760 + 0.649i)7-s + (−0.235 + 0.113i)8-s + (−0.177 − 0.0854i)10-s + (0.0966 + 0.121i)11-s + (−0.0924 − 0.115i)13-s + (1.31 + 0.255i)14-s + (1.04 + 0.501i)16-s + (0.426 + 1.86i)17-s − 1.16·19-s + (0.0262 + 0.115i)20-s + (0.0463 − 0.203i)22-s + (0.00915 − 0.0401i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.800 - 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.500304 + 0.166759i\)
\(L(\frac12)\) \(\approx\) \(0.500304 + 0.166759i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.01 - 1.71i)T \)
good2 \( 1 + (1.18 + 1.48i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (-0.295 + 0.142i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.320 - 0.402i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.333 + 0.417i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.75 - 7.70i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + 5.08T + 19T^{2} \)
23 \( 1 + (-0.0438 + 0.192i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.477 - 2.09i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 2.48T + 31T^{2} \)
37 \( 1 + (-1.18 - 5.19i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-1.92 + 0.925i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-9.46 - 4.55i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (5.00 + 6.27i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (1.18 - 5.19i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (-0.263 - 0.127i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-0.889 - 3.89i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + (-2.09 + 9.15i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (6.25 - 7.83i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 - 3.83T + 79T^{2} \)
83 \( 1 + (-5.28 + 6.63i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (8.30 - 10.4i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 9.62T + 97T^{2} \)
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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

−10.96776392742006064486893663733, −10.31622058578392160059240484255, −9.550602078243182225805211937690, −8.759225705617361951269097998152, −8.003152599231915045127698917837, −6.42452316620536208755208523841, −5.69108256600870302639172527053, −3.95349383735327145358148550244, −2.79019965848482162905048931875, −1.61234711747438839892130363858, 0.43435468915178615116502305146, 2.83118697072616163217895633253, 4.26178455338563188764748238456, 5.74436470848593390776858245678, 6.60667858078212171773251696175, 7.31722322571627112469484060355, 8.153514792236739335027065578248, 9.278625056659135810998612656714, 9.737856867783712216519127175931, 10.72133118133018622278392203963

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