Properties

Label 2-21e2-147.104-c1-0-11
Degree $2$
Conductor $441$
Sign $-0.412 - 0.911i$
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  + (2.13 + 1.70i)2-s + (1.21 + 5.30i)4-s + (1.31 + 0.635i)5-s + (−0.795 − 2.52i)7-s + (−4.07 + 8.46i)8-s + (1.73 + 3.60i)10-s + (−2.75 − 2.19i)11-s + (4.51 + 3.60i)13-s + (2.59 − 6.73i)14-s + (−13.2 + 6.40i)16-s + (−0.0162 + 0.0710i)17-s − 4.52i·19-s + (−1.77 + 7.77i)20-s + (−2.14 − 9.37i)22-s + (−1.76 + 0.402i)23-s + ⋯
L(s)  = 1  + (1.50 + 1.20i)2-s + (0.605 + 2.65i)4-s + (0.590 + 0.284i)5-s + (−0.300 − 0.953i)7-s + (−1.44 + 2.99i)8-s + (0.548 + 1.13i)10-s + (−0.831 − 0.662i)11-s + (1.25 + 0.999i)13-s + (0.693 − 1.80i)14-s + (−3.32 + 1.60i)16-s + (−0.00393 + 0.0172i)17-s − 1.03i·19-s + (−0.396 + 1.73i)20-s + (−0.456 − 1.99i)22-s + (−0.367 + 0.0839i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.412 - 0.911i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.412 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71008 + 2.65122i\)
\(L(\frac12)\) \(\approx\) \(1.71008 + 2.65122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.795 + 2.52i)T \)
good2 \( 1 + (-2.13 - 1.70i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (-1.31 - 0.635i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (2.75 + 2.19i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-4.51 - 3.60i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.0162 - 0.0710i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 4.52iT - 19T^{2} \)
23 \( 1 + (1.76 - 0.402i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-2.61 - 0.597i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 7.13iT - 31T^{2} \)
37 \( 1 + (-1.33 + 5.82i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-11.3 - 5.46i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (1.31 - 0.632i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (5.84 - 7.32i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (9.10 - 2.07i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (0.107 - 0.0516i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-3.20 - 0.730i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 5.78T + 67T^{2} \)
71 \( 1 + (-0.0872 + 0.0199i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.43 + 2.73i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 + (-0.251 - 0.315i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (6.88 + 8.63i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 14.7iT - 97T^{2} \)
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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

−11.44908451163882889965592213825, −10.92418931710411867710446842299, −9.426496414843661319948873228928, −8.221622049011620087862937857990, −7.42903657017561554912719589425, −6.33881428345163554693661984474, −6.03336001244112379057325067606, −4.66371647091511517445911059057, −3.83298574158990682105040015691, −2.66932098525795984151342634890, 1.58730817367291572626856005101, 2.71012023154122780340548383001, 3.70621002012798380015739635581, 5.10616357334407096419542486265, 5.64470100237929825255925976198, 6.44853586740694630198948570998, 8.257355797930044225514978323178, 9.543900720965425581790947459656, 10.21471264790572752426095664824, 10.96182732254355597819561279946

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