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Properties

Label 2-21e2-63.4-c1-0-14
Degree $2$
Conductor $441$
Sign $0.386 - 0.922i$
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  + (−1.26 + 2.19i)2-s + (1.70 − 0.300i)3-s + (−2.20 − 3.82i)4-s − 0.879·5-s + (−1.5 + 4.12i)6-s + 6.10·8-s + (2.81 − 1.02i)9-s + (1.11 − 1.92i)10-s + 3.87·11-s + (−4.91 − 5.85i)12-s + (2.72 − 4.72i)13-s + (−1.49 + 0.264i)15-s + (−3.31 + 5.74i)16-s + (−0.826 + 1.43i)17-s + (−1.31 + 7.48i)18-s + (−1.20 − 2.08i)19-s + ⋯
L(s)  = 1  + (−0.895 + 1.55i)2-s + (0.984 − 0.173i)3-s + (−1.10 − 1.91i)4-s − 0.393·5-s + (−0.612 + 1.68i)6-s + 2.15·8-s + (0.939 − 0.342i)9-s + (0.352 − 0.609i)10-s + 1.16·11-s + (−1.41 − 1.68i)12-s + (0.756 − 1.30i)13-s + (−0.387 + 0.0682i)15-s + (−0.829 + 1.43i)16-s + (−0.200 + 0.347i)17-s + (−0.310 + 1.76i)18-s + (−0.276 − 0.479i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
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: \((2,\ 441,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.989132 + 0.657953i\)
\(L(\frac12)\) \(\approx\) \(0.989132 + 0.657953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.300i)T \)
7 \( 1 \)
good2 \( 1 + (1.26 - 2.19i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 0.879T + 5T^{2} \)
11 \( 1 - 3.87T + 11T^{2} \)
13 \( 1 + (-2.72 + 4.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.826 - 1.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.20 + 2.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.16T + 23T^{2} \)
29 \( 1 + (-3.02 - 5.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.27 - 3.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.27 - 3.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.592 + 1.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0923 + 0.160i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.511 + 0.885i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.64 - 6.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.33 + 5.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.29 + 2.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.47 - 2.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 + (-6.39 + 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.97 + 5.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.109 - 0.189i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.51 + 9.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.25 + 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.88919837462206476515977179858, −9.942660681219486117946263306610, −9.004378846444553257152230690407, −8.492564635350077131706833314733, −7.75330638895902582505778299944, −6.86244500321361413650105875695, −6.11140711525917726190799891190, −4.70079234571926891255859011501, −3.36873537750804530837581543237, −1.16988591322485282112455651317, 1.38781723698542096110292637509, 2.50513848106726247676398584109, 3.86459180547307648357347082800, 4.19676205139258560891287878343, 6.60632671970344356580981990757, 7.83498905910377932038268695488, 8.607515851812954866572052428020, 9.330609183293253550038182261347, 9.820870921977605294554954910621, 11.02254309381365829901519717328

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