Properties

Label 2-21e2-441.142-c1-0-9
Degree $2$
Conductor $441$
Sign $-0.292 + 0.956i$
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  + (−0.723 + 1.84i)2-s + (0.680 + 1.59i)3-s + (−1.40 − 1.30i)4-s + (−3.94 + 1.90i)5-s + (−3.42 + 0.101i)6-s + (2.61 + 0.408i)7-s + (−0.143 + 0.0693i)8-s + (−2.07 + 2.16i)9-s + (−0.648 − 8.65i)10-s + (2.48 + 3.11i)11-s + (1.12 − 3.12i)12-s + (−0.678 + 1.72i)13-s + (−2.64 + 4.52i)14-s + (−5.71 − 4.99i)15-s + (−0.310 − 4.14i)16-s + (2.33 − 2.16i)17-s + ⋯
L(s)  = 1  + (−0.511 + 1.30i)2-s + (0.392 + 0.919i)3-s + (−0.703 − 0.652i)4-s + (−1.76 + 0.850i)5-s + (−1.39 + 0.0413i)6-s + (0.988 + 0.154i)7-s + (−0.0508 + 0.0245i)8-s + (−0.691 + 0.722i)9-s + (−0.204 − 2.73i)10-s + (0.749 + 0.940i)11-s + (0.324 − 0.903i)12-s + (−0.188 + 0.479i)13-s + (−0.706 + 1.20i)14-s + (−1.47 − 1.28i)15-s + (−0.0776 − 1.03i)16-s + (0.566 − 0.525i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.292 + 0.956i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.292 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.463477 - 0.626514i\)
\(L(\frac12)\) \(\approx\) \(0.463477 - 0.626514i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.680 - 1.59i)T \)
7 \( 1 + (-2.61 - 0.408i)T \)
good2 \( 1 + (0.723 - 1.84i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (3.94 - 1.90i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-2.48 - 3.11i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.678 - 1.72i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-2.33 + 2.16i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.202 - 0.350i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.50 + 6.58i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.69 + 0.832i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (3.16 - 5.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.63 - 1.73i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-3.76 - 2.56i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (6.79 - 4.63i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (2.65 - 6.75i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (3.75 - 1.15i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-2.71 + 1.84i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (2.89 - 2.68i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-0.467 + 0.808i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.209 + 0.918i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (2.49 + 0.376i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-2.76 - 4.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.437 + 1.11i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (4.08 + 10.4i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (4.31 - 7.48i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.55524210330465298449774479791, −10.86130022391740491595699589260, −9.668377310590879307401931418130, −8.724729265448433345040019240187, −7.983441895811996229215955824629, −7.40898996715686215641052265037, −6.52691104250052448087498933763, −4.88785716252023858959040122366, −4.25097716919748057542821048578, −2.89779594310863030119237954056, 0.59681514759461082272037780038, 1.56855560719768128769264194190, 3.33939204486170859357490352756, 3.96626796917058599840609880238, 5.58524675753933882193038347374, 7.28396355277804509954583148485, 8.067138677672280584667409891739, 8.584651002177227606133578634945, 9.410426152987224463991832421725, 11.04976471721558107180592118615

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