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Properties

Label 2-21e2-63.20-c1-0-24
Degree $2$
Conductor $441$
Sign $0.974 + 0.222i$
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  + (0.575 + 0.332i)2-s + (1.69 + 0.358i)3-s + (−0.779 − 1.34i)4-s + (−0.0141 − 0.0245i)5-s + (0.855 + 0.768i)6-s − 2.36i·8-s + (2.74 + 1.21i)9-s − 0.0188i·10-s + (−0.885 − 0.511i)11-s + (−0.837 − 2.56i)12-s + (4.87 − 2.81i)13-s + (−0.0152 − 0.0466i)15-s + (−0.773 + 1.33i)16-s + 5.67·17-s + (1.17 + 1.60i)18-s + 2.09i·19-s + ⋯
L(s)  = 1  + (0.406 + 0.234i)2-s + (0.978 + 0.206i)3-s + (−0.389 − 0.674i)4-s + (−0.00632 − 0.0109i)5-s + (0.349 + 0.313i)6-s − 0.835i·8-s + (0.914 + 0.404i)9-s − 0.00594i·10-s + (−0.266 − 0.154i)11-s + (−0.241 − 0.740i)12-s + (1.35 − 0.781i)13-s + (−0.00392 − 0.0120i)15-s + (−0.193 + 0.334i)16-s + 1.37·17-s + (0.276 + 0.379i)18-s + 0.480i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.974 + 0.222i$
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: \((2,\ 441,\ (\ :1/2),\ 0.974 + 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20352 - 0.248596i\)
\(L(\frac12)\) \(\approx\) \(2.20352 - 0.248596i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 - 0.358i)T \)
7 \( 1 \)
good2 \( 1 + (-0.575 - 0.332i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.0141 + 0.0245i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.885 + 0.511i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.87 + 2.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.67T + 17T^{2} \)
19 \( 1 - 2.09iT - 19T^{2} \)
23 \( 1 + (6.28 - 3.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.52 + 2.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.87 - 1.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 + (3.52 + 6.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.15 - 2.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.43 - 9.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 + (-3.01 - 5.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.05 + 1.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.38 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.93iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 + (-7.80 + 13.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.07 - 5.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (6.77 + 3.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.71847851809024879078121421939, −10.16723682449232316673531218512, −9.313844917296943520433490880498, −8.330478005177413964947698324161, −7.60249185394330158481695099831, −6.15333014622114322796487201703, −5.40449826629025672162593929702, −4.07925126605797660134613285741, −3.28122966582167861469615653681, −1.43578997257656420505702659554, 1.86835847824630152839813055842, 3.30184981183990256783221388520, 3.90165056340480995000917562177, 5.17536250650857044078049742940, 6.63080919703763732970650851456, 7.70939331278430746112108854504, 8.423970178741125574788232278888, 9.144412182022515565050235618484, 10.15351806313989446881231579173, 11.37571327104534432419419462069

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