Properties

Label 2-21e2-147.41-c1-0-2
Degree $2$
Conductor $441$
Sign $-0.917 - 0.398i$
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  + (−1.64 + 1.31i)2-s + (0.542 − 2.37i)4-s + (0.0408 − 0.0196i)5-s + (0.337 − 2.62i)7-s + (0.399 + 0.829i)8-s + (−0.0413 + 0.0859i)10-s + (−3.28 + 2.62i)11-s + (−1.87 + 1.49i)13-s + (2.89 + 4.76i)14-s + (2.64 + 1.27i)16-s + (1.21 + 5.30i)17-s + 3.24i·19-s + (−0.0245 − 0.107i)20-s + (1.97 − 8.63i)22-s + (3.85 + 0.880i)23-s + ⋯
L(s)  = 1  + (−1.16 + 0.928i)2-s + (0.271 − 1.18i)4-s + (0.0182 − 0.00878i)5-s + (0.127 − 0.991i)7-s + (0.141 + 0.293i)8-s + (−0.0130 + 0.0271i)10-s + (−0.991 + 0.790i)11-s + (−0.520 + 0.414i)13-s + (0.772 + 1.27i)14-s + (0.661 + 0.318i)16-s + (0.293 + 1.28i)17-s + 0.743i·19-s + (−0.00549 − 0.0240i)20-s + (0.420 − 1.84i)22-s + (0.804 + 0.183i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.917 - 0.398i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.917 - 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0925595 + 0.445779i\)
\(L(\frac12)\) \(\approx\) \(0.0925595 + 0.445779i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.337 + 2.62i)T \)
good2 \( 1 + (1.64 - 1.31i)T + (0.445 - 1.94i)T^{2} \)
5 \( 1 + (-0.0408 + 0.0196i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (3.28 - 2.62i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.87 - 1.49i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-1.21 - 5.30i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 - 3.24iT - 19T^{2} \)
23 \( 1 + (-3.85 - 0.880i)T + (20.7 + 9.97i)T^{2} \)
29 \( 1 + (4.40 - 1.00i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 - 5.18iT - 31T^{2} \)
37 \( 1 + (-0.882 - 3.86i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (-4.08 + 1.96i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-4.96 - 2.39i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (5.79 + 7.27i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (5.38 + 1.22i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 + (-3.45 - 1.66i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (12.2 - 2.79i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 - 3.28T + 67T^{2} \)
71 \( 1 + (-1.72 - 0.393i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-8.05 - 6.42i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + 6.50T + 79T^{2} \)
83 \( 1 + (-2.63 + 3.29i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-6.50 + 8.16i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 18.6iT - 97T^{2} \)
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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

−11.08029444675352937077677232159, −10.23983849205361299277627171130, −9.743006153020506638196071444415, −8.645336979543256119133632004049, −7.64573270473232785062232954520, −7.33171696447371513504060868318, −6.21362328084125178035994455805, −5.03199779190496344385989378948, −3.66041745049917544893403891625, −1.57853786062773059481077378311, 0.41890568098200968635478340277, 2.36678621203802016743559609407, 2.98728962108034414427575032158, 4.99317572652140710483234254015, 5.89809818991743810086142922070, 7.55765974675802103465638215258, 8.171588172697076805099492855325, 9.250182020200499075887056335169, 9.621968670457389339507442593306, 10.86730117159911375979956295188

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