Properties

Label 2-21e2-441.131-c1-0-17
Degree $2$
Conductor $441$
Sign $0.862 + 0.506i$
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  + (−0.419 − 0.334i)2-s + (−1.08 + 1.35i)3-s + (−0.380 − 1.66i)4-s + (0.121 + 1.62i)5-s + (0.906 − 0.205i)6-s + (−2.56 − 0.655i)7-s + (−0.864 + 1.79i)8-s + (−0.661 − 2.92i)9-s + (0.491 − 0.721i)10-s + (1.54 − 0.605i)11-s + (2.67 + 1.28i)12-s + (5.17 − 2.03i)13-s + (0.856 + 1.13i)14-s + (−2.32 − 1.58i)15-s + (−2.12 + 1.02i)16-s + (4.27 − 1.31i)17-s + ⋯
L(s)  = 1  + (−0.296 − 0.236i)2-s + (−0.624 + 0.781i)3-s + (−0.190 − 0.834i)4-s + (0.0543 + 0.725i)5-s + (0.370 − 0.0840i)6-s + (−0.968 − 0.247i)7-s + (−0.305 + 0.634i)8-s + (−0.220 − 0.975i)9-s + (0.155 − 0.228i)10-s + (0.465 − 0.182i)11-s + (0.770 + 0.372i)12-s + (1.43 − 0.563i)13-s + (0.228 + 0.302i)14-s + (−0.600 − 0.410i)15-s + (−0.530 + 0.255i)16-s + (1.03 − 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.862 + 0.506i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.862 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.817990 - 0.222533i\)
\(L(\frac12)\) \(\approx\) \(0.817990 - 0.222533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08 - 1.35i)T \)
7 \( 1 + (2.56 + 0.655i)T \)
good2 \( 1 + (0.419 + 0.334i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (-0.121 - 1.62i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-1.54 + 0.605i)T + (8.06 - 7.48i)T^{2} \)
13 \( 1 + (-5.17 + 2.03i)T + (9.52 - 8.84i)T^{2} \)
17 \( 1 + (-4.27 + 1.31i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (0.895 + 0.516i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.69 - 2.90i)T + (-1.71 + 22.9i)T^{2} \)
29 \( 1 + (1.15 + 3.74i)T + (-23.9 + 16.3i)T^{2} \)
31 \( 1 + 2.17iT - 31T^{2} \)
37 \( 1 + (4.26 + 3.95i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-9.00 + 6.14i)T + (14.9 - 38.1i)T^{2} \)
43 \( 1 + (-3.00 - 2.04i)T + (15.7 + 40.0i)T^{2} \)
47 \( 1 + (-2.85 + 3.58i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.67 - 1.80i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (-7.69 + 3.70i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (7.35 + 1.67i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 8.09T + 67T^{2} \)
71 \( 1 + (-0.590 + 0.134i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (2.26 + 0.888i)T + (53.5 + 49.6i)T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 + (5.28 - 13.4i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (10.7 - 1.62i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-14.8 + 8.57i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.83556496578053832551020205030, −10.27291986765121201527268866355, −9.490237246922075774218475782039, −8.769239924945687695432286258064, −7.10297184151892068013107640946, −6.03914927058970094890814012688, −5.63738954335137170763634643714, −4.04468101951274178620925170898, −3.07204539914901785686421562053, −0.817056095579916080108247300060, 1.17100952787982900141022855297, 3.12827483547523435750732514404, 4.38318303929796388897666345259, 5.80894281056171876362619327040, 6.58648045113335324877862125613, 7.43878665745688911036690606582, 8.640578689944981153375906185788, 8.965156064308530910092595171210, 10.27913406593298097193139725185, 11.43431660577355581922867306291

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