Properties

Label 2-21e2-441.142-c1-0-49
Degree $2$
Conductor $441$
Sign $-0.949 + 0.312i$
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  + (0.617 − 1.57i)2-s + (−0.0875 − 1.72i)3-s + (−0.625 − 0.580i)4-s + (2.02 − 0.972i)5-s + (−2.77 − 0.929i)6-s + (−2.54 − 0.730i)7-s + (1.74 − 0.840i)8-s + (−2.98 + 0.302i)9-s + (−0.283 − 3.77i)10-s + (0.293 + 0.367i)11-s + (−0.949 + 1.13i)12-s + (−0.290 + 0.739i)13-s + (−2.71 + 3.54i)14-s + (−1.85 − 3.40i)15-s + (−0.372 − 4.96i)16-s + (−0.101 + 0.0942i)17-s + ⋯
L(s)  = 1  + (0.436 − 1.11i)2-s + (−0.0505 − 0.998i)3-s + (−0.312 − 0.290i)4-s + (0.903 − 0.435i)5-s + (−1.13 − 0.379i)6-s + (−0.961 − 0.276i)7-s + (0.616 − 0.297i)8-s + (−0.994 + 0.100i)9-s + (−0.0895 − 1.19i)10-s + (0.0884 + 0.110i)11-s + (−0.274 + 0.327i)12-s + (−0.0804 + 0.205i)13-s + (−0.726 + 0.948i)14-s + (−0.480 − 0.880i)15-s + (−0.0930 − 1.24i)16-s + (−0.0246 + 0.0228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.949 + 0.312i$
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: \((2,\ 441,\ (\ :1/2),\ -0.949 + 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.295172 - 1.84189i\)
\(L(\frac12)\) \(\approx\) \(0.295172 - 1.84189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0875 + 1.72i)T \)
7 \( 1 + (2.54 + 0.730i)T \)
good2 \( 1 + (-0.617 + 1.57i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-2.02 + 0.972i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.293 - 0.367i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.290 - 0.739i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (0.101 - 0.0942i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-2.14 + 3.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.596 - 2.61i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.30 - 0.403i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-0.681 + 1.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.82 - 3.03i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-4.87 - 3.32i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (5.40 - 3.68i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-1.41 + 3.59i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (13.5 - 4.17i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-5.05 + 3.44i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-2.52 + 2.34i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-2.23 + 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.541 - 2.37i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.44 - 0.821i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (4.19 + 7.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.705 - 1.79i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-5.35 - 13.6i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-4.20 + 7.28i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.01359730298722891920093994218, −9.792862604772775362668144781113, −9.336455664251119707706411417127, −7.86592381200114595243535627985, −6.87476322604204078247418624279, −6.00728323255464264234465903967, −4.78079620001700063416738910601, −3.29647009867491751273748787687, −2.32341581088540812835475429872, −1.10640623208017843603728087274, 2.55095899632004547191773185979, 3.86831393300604193852034432803, 5.12089985015724803199789129527, 6.01468655588446979184497930917, 6.39072829078165567457596144887, 7.71788379779036693447292796854, 8.853227214575536161742674902929, 9.852846627092209138853689551004, 10.31243223928444858308115488941, 11.34519045978633210455218973344

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