Properties

Label 2-21e2-147.104-c1-0-10
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

Related objects

Downloads

Learn more

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: $\chi_{441} (251, \cdot )$
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 \( 1 - 18.6iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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