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Properties

Label 2-21e2-63.41-c1-0-7
Degree $2$
Conductor $441$
Sign $-0.129 - 0.991i$
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  + (1.61 − 0.934i)2-s + (−1.01 + 1.40i)3-s + (0.746 − 1.29i)4-s + (−1.25 + 2.17i)5-s + (−0.326 + 3.22i)6-s + 0.947i·8-s + (−0.948 − 2.84i)9-s + 4.68i·10-s + (−4.85 + 2.80i)11-s + (1.06 + 2.35i)12-s + (−0.384 − 0.221i)13-s + (−1.78 − 3.95i)15-s + (2.37 + 4.11i)16-s − 3.07·17-s + (−4.19 − 3.71i)18-s − 2.57i·19-s + ⋯
L(s)  = 1  + (1.14 − 0.660i)2-s + (−0.584 + 0.811i)3-s + (0.373 − 0.646i)4-s + (−0.560 + 0.970i)5-s + (−0.133 + 1.31i)6-s + 0.335i·8-s + (−0.316 − 0.948i)9-s + 1.48i·10-s + (−1.46 + 0.845i)11-s + (0.306 + 0.680i)12-s + (−0.106 − 0.0615i)13-s + (−0.459 − 1.02i)15-s + (0.594 + 1.02i)16-s − 0.746·17-s + (−0.988 − 0.876i)18-s − 0.590i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.129 - 0.991i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.129 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.933154 + 1.06302i\)
\(L(\frac12)\) \(\approx\) \(0.933154 + 1.06302i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.01 - 1.40i)T \)
7 \( 1 \)
good2 \( 1 + (-1.61 + 0.934i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.25 - 2.17i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.85 - 2.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.384 + 0.221i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.07T + 17T^{2} \)
19 \( 1 + 2.57iT - 19T^{2} \)
23 \( 1 + (-6.83 - 3.94i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.71 + 1.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9.06 - 5.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 + (1.64 - 2.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.75 + 8.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.07 - 1.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.85iT - 53T^{2} \)
59 \( 1 + (-3.65 + 6.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.40 - 4.27i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.934 + 1.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.95iT - 71T^{2} \)
73 \( 1 + 8.51iT - 73T^{2} \)
79 \( 1 + (-0.287 - 0.497i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.23 + 7.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.57T + 89T^{2} \)
97 \( 1 + (-3.22 + 1.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.39125911702197929297896632390, −10.67614874883785292855598955599, −10.18164426873788984524058293206, −8.782632783476845234780304424114, −7.46507028629814430110638353149, −6.46187246754893372143562204666, −5.08587810416726139012673068809, −4.68863189990324504504652115795, −3.38245350348683927223859249867, −2.65307095692928756927716342213, 0.65563703019163055041713823978, 2.86204872663296284953253741345, 4.53595754955647861008905981802, 5.08410524257717121972457355974, 6.01575876783654914975243268037, 6.90332883906542028371792452898, 7.971106337326193416156046320970, 8.563382582752426169008379640183, 10.20735159162746399448882519445, 11.21563052470888860247757661074

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