Properties

Label 2-21e2-63.4-c1-0-0
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: $\chi_{441} (67, \cdot )$
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

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

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