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Properties

Label 2-21e2-49.22-c1-0-8
Degree $2$
Conductor $441$
Sign $0.183 - 0.983i$
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  + (−1.54 + 0.744i)2-s + (0.590 − 0.740i)4-s + (−0.580 + 2.54i)5-s + (2.64 − 0.137i)7-s + (0.402 − 1.76i)8-s + (−0.996 − 4.36i)10-s + (4.78 − 2.30i)11-s + (2.79 − 1.34i)13-s + (−3.98 + 2.18i)14-s + (1.11 + 4.87i)16-s + (1.56 + 1.96i)17-s − 7.74·19-s + (1.53 + 1.93i)20-s + (−5.68 + 7.12i)22-s + (1.61 − 2.02i)23-s + ⋯
L(s)  = 1  + (−1.09 + 0.526i)2-s + (0.295 − 0.370i)4-s + (−0.259 + 1.13i)5-s + (0.998 − 0.0519i)7-s + (0.142 − 0.623i)8-s + (−0.315 − 1.38i)10-s + (1.44 − 0.694i)11-s + (0.776 − 0.373i)13-s + (−1.06 + 0.582i)14-s + (0.277 + 1.21i)16-s + (0.379 + 0.476i)17-s − 1.77·19-s + (0.344 + 0.431i)20-s + (−1.21 + 1.51i)22-s + (0.337 − 0.422i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.183 - 0.983i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.183 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.680723 + 0.565411i\)
\(L(\frac12)\) \(\approx\) \(0.680723 + 0.565411i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.137i)T \)
good2 \( 1 + (1.54 - 0.744i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (0.580 - 2.54i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-4.78 + 2.30i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-2.79 + 1.34i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-1.56 - 1.96i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + 7.74T + 19T^{2} \)
23 \( 1 + (-1.61 + 2.02i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-3.80 - 4.76i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 2.25T + 31T^{2} \)
37 \( 1 + (-6.33 - 7.94i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-0.261 + 1.14i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (0.464 + 2.03i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (10.4 - 5.02i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (5.52 - 6.92i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.55 + 6.80i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (3.28 + 4.11i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 3.66T + 67T^{2} \)
71 \( 1 + (0.0227 - 0.0285i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (4.18 + 2.01i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 + (-5.75 - 2.77i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-7.45 - 3.59i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + 1.62T + 97T^{2} \)
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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

−10.76962153224044243829384368709, −10.70282022550131607241430214158, −9.284376563078713646287959248246, −8.396799333929277427902815130774, −7.962807826301499147808246368543, −6.58522811274396004460995324556, −6.37347420996797121217576626303, −4.38838811871538868834455674375, −3.31125765191733498293707591179, −1.33011467115463539333003125059, 1.03283085272075135107605424482, 1.97523499882492676139629437426, 4.18275985105466946546888983555, 4.87403604200862855747741620917, 6.31670533598238096170191331664, 7.71476432871564951181677208537, 8.547578262561203483288535961307, 8.998590874718244788542771652570, 9.837802822200217948752265583053, 10.96769210106545229756099334739

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