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Properties

Label 2-21e2-63.16-c1-0-28
Degree $2$
Conductor $441$
Sign $-0.841 - 0.539i$
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  + (−1.08 − 1.88i)2-s + (1.18 + 1.26i)3-s + (−1.36 + 2.36i)4-s − 1.26·5-s + (1.10 − 3.60i)6-s + 1.60·8-s + (−0.213 + 2.99i)9-s + (1.38 + 2.39i)10-s − 5.47·11-s + (−4.61 + 1.06i)12-s + (−2.37 − 4.10i)13-s + (−1.49 − 1.60i)15-s + (0.992 + 1.71i)16-s + (−2.40 − 4.17i)17-s + (5.87 − 2.85i)18-s + (2.69 − 4.66i)19-s + ⋯
L(s)  = 1  + (−0.769 − 1.33i)2-s + (0.681 + 0.731i)3-s + (−0.684 + 1.18i)4-s − 0.567·5-s + (0.450 − 1.47i)6-s + 0.566·8-s + (−0.0710 + 0.997i)9-s + (0.436 + 0.755i)10-s − 1.65·11-s + (−1.33 + 0.306i)12-s + (−0.658 − 1.13i)13-s + (−0.386 − 0.415i)15-s + (0.248 + 0.429i)16-s + (−0.584 − 1.01i)17-s + (1.38 − 0.672i)18-s + (0.617 − 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.841 - 0.539i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.841 - 0.539i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0555312 + 0.189597i\)
\(L(\frac12)\) \(\approx\) \(0.0555312 + 0.189597i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.18 - 1.26i)T \)
7 \( 1 \)
good2 \( 1 + (1.08 + 1.88i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 1.26T + 5T^{2} \)
11 \( 1 + 5.47T + 11T^{2} \)
13 \( 1 + (2.37 + 4.10i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.40 + 4.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.17T + 23T^{2} \)
29 \( 1 + (-2.01 + 3.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.732 - 1.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.959 - 1.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.94 - 3.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.66 - 2.87i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.57 - 2.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.57 - 6.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.154 + 0.267i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.17 - 8.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.23 - 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.96T + 71T^{2} \)
73 \( 1 + (5.27 + 9.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.50 - 7.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.08 + 8.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.59 + 4.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.48 + 4.30i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.40500618143146838006942173156, −9.930672261431341593609587757424, −9.056355148088723743343456663541, −8.033095658141106945919568670441, −7.59816636843226012869813768243, −5.40698096643441104034185113941, −4.35590337718280638481807306967, −2.96426106303528574809993630830, −2.53329642325872098174831785024, −0.13536677525438883948570283726, 2.14508860689381345644091074641, 3.80834502277277001868465066683, 5.38969986150021017236289412565, 6.42316488094206142565525294431, 7.33973038700976550336645236774, 7.939320117608654112005909752155, 8.477813295339845384726246927667, 9.511979699134632676807126880942, 10.36257176727424831943175137080, 11.82898665379609895249505301792

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