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Properties

Label 2-21e2-441.320-c1-0-11
Degree $2$
Conductor $441$
Sign $0.839 + 0.543i$
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  + (−1.05 − 2.19i)2-s + (1.37 + 1.05i)3-s + (−2.44 + 3.06i)4-s + (−3.22 + 0.994i)5-s + (0.871 − 4.12i)6-s + (−0.531 − 2.59i)7-s + (4.56 + 1.04i)8-s + (0.761 + 2.90i)9-s + (5.58 + 6.02i)10-s + (2.38 + 0.178i)11-s + (−6.59 + 1.61i)12-s + (5.81 + 0.435i)13-s + (−5.12 + 3.90i)14-s + (−5.47 − 2.04i)15-s + (−0.789 − 3.45i)16-s + (−1.47 + 3.74i)17-s + ⋯
L(s)  = 1  + (−0.746 − 1.55i)2-s + (0.791 + 0.610i)3-s + (−1.22 + 1.53i)4-s + (−1.44 + 0.444i)5-s + (0.355 − 1.68i)6-s + (−0.200 − 0.979i)7-s + (1.61 + 0.368i)8-s + (0.253 + 0.967i)9-s + (1.76 + 1.90i)10-s + (0.720 + 0.0539i)11-s + (−1.90 + 0.467i)12-s + (1.61 + 0.120i)13-s + (−1.36 + 1.04i)14-s + (−1.41 − 0.528i)15-s + (−0.197 − 0.864i)16-s + (−0.356 + 0.908i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.839 + 0.543i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (320, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.839 + 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.890862 - 0.263433i\)
\(L(\frac12)\) \(\approx\) \(0.890862 - 0.263433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.37 - 1.05i)T \)
7 \( 1 + (0.531 + 2.59i)T \)
good2 \( 1 + (1.05 + 2.19i)T + (-1.24 + 1.56i)T^{2} \)
5 \( 1 + (3.22 - 0.994i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (-2.38 - 0.178i)T + (10.8 + 1.63i)T^{2} \)
13 \( 1 + (-5.81 - 0.435i)T + (12.8 + 1.93i)T^{2} \)
17 \( 1 + (1.47 - 3.74i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (-3.32 - 1.92i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.156 + 1.03i)T + (-21.9 - 6.77i)T^{2} \)
29 \( 1 + (-5.17 - 2.02i)T + (21.2 + 19.7i)T^{2} \)
31 \( 1 - 2.94iT - 31T^{2} \)
37 \( 1 + (-11.0 + 1.65i)T + (35.3 - 10.9i)T^{2} \)
41 \( 1 + (3.77 + 3.50i)T + (3.06 + 40.8i)T^{2} \)
43 \( 1 + (2.94 - 2.73i)T + (3.21 - 42.8i)T^{2} \)
47 \( 1 + (2.68 - 1.29i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-1.05 + 7.02i)T + (-50.6 - 15.6i)T^{2} \)
59 \( 1 + (0.112 + 0.493i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (5.84 - 4.65i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + 1.32T + 67T^{2} \)
71 \( 1 + (-8.15 - 6.50i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (-9.25 + 0.693i)T + (72.1 - 10.8i)T^{2} \)
79 \( 1 + 6.76T + 79T^{2} \)
83 \( 1 + (-1.27 - 17.0i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (4.11 - 2.80i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 + (4.19 - 2.42i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.98696124740679669933008857151, −10.33408354352425068410354420368, −9.422488274334711155989184349769, −8.388687805824409731418922759273, −8.031526497529015735017363415471, −6.73863967126059369168432322333, −4.22789105205719577524150107315, −3.80842687127078923762820885169, −3.11663546755621013920314262771, −1.27790858468826200793248452721, 0.867993191325580831737400371075, 3.24126062472131647148310986474, 4.57729552144606486044886503164, 6.01834247697709697530241920167, 6.75641798135345446882619597039, 7.76517449458812545923241791504, 8.307521090412450153593726047613, 8.965003591236453422347457454992, 9.538988644228209387364703157978, 11.43871408003442400361532528307

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