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Properties

Label 2-21e2-63.20-c1-0-18
Degree $2$
Conductor $441$
Sign $0.660 + 0.751i$
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  + (−0.367 − 0.212i)2-s + (1.08 − 1.34i)3-s + (−0.910 − 1.57i)4-s + (1.80 + 3.12i)5-s + (−0.685 + 0.265i)6-s + 1.62i·8-s + (−0.639 − 2.93i)9-s − 1.53i·10-s + (3.20 + 1.85i)11-s + (−3.11 − 0.484i)12-s + (5.23 − 3.02i)13-s + (6.17 + 0.960i)15-s + (−1.47 + 2.55i)16-s − 1.06·17-s + (−0.386 + 1.21i)18-s − 3.65i·19-s + ⋯
L(s)  = 1  + (−0.259 − 0.149i)2-s + (0.627 − 0.778i)3-s + (−0.455 − 0.788i)4-s + (0.806 + 1.39i)5-s + (−0.279 + 0.108i)6-s + 0.572i·8-s + (−0.213 − 0.977i)9-s − 0.483i·10-s + (0.967 + 0.558i)11-s + (−0.899 − 0.139i)12-s + (1.45 − 0.838i)13-s + (1.59 + 0.248i)15-s + (−0.369 + 0.639i)16-s − 0.258·17-s + (−0.0911 + 0.285i)18-s − 0.837i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.660 + 0.751i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.660 + 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48223 - 0.670593i\)
\(L(\frac12)\) \(\approx\) \(1.48223 - 0.670593i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.08 + 1.34i)T \)
7 \( 1 \)
good2 \( 1 + (0.367 + 0.212i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.80 - 3.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.20 - 1.85i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.23 + 3.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.06T + 17T^{2} \)
19 \( 1 + 3.65iT - 19T^{2} \)
23 \( 1 + (-0.314 + 0.181i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.857 + 0.495i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.939 + 0.542i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.00T + 37T^{2} \)
41 \( 1 + (2.09 + 3.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.89 - 3.28i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.83 + 4.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.53iT - 53T^{2} \)
59 \( 1 + (-5.62 - 9.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0238 - 0.0137i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.86 - 8.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.55iT - 71T^{2} \)
73 \( 1 - 2.25iT - 73T^{2} \)
79 \( 1 + (3.26 - 5.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.52 + 2.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14.9T + 89T^{2} \)
97 \( 1 + (-1.67 - 0.964i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.81402393433252039037050456560, −10.10463576442982251142447587648, −9.188714355662122970949906987859, −8.494579506207555482987878908687, −7.08245082957037915341896465195, −6.45769156770864271585104851946, −5.62449403835305121128313281810, −3.78505353235110188429599924538, −2.52975355605410414684448700316, −1.37041541248543645620429212287, 1.57921116896323025731548073805, 3.54877782889569672310453729208, 4.24002268853988859723270727523, 5.33884828169998565982295034108, 6.54338724968595810393500461856, 8.170464937028931144234055351508, 8.702959019690244269900046629049, 9.157375975664855842342126541452, 9.913270927196266441588712730983, 11.21125458401159776917345512272

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