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Properties

Label 2-21e2-49.36-c1-0-7
Degree $2$
Conductor $441$
Sign $0.715 - 0.699i$
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  + (0.132 + 0.580i)2-s + (1.48 − 0.714i)4-s + (−0.595 − 0.747i)5-s + (−0.416 + 2.61i)7-s + (1.35 + 1.69i)8-s + (0.354 − 0.444i)10-s + (0.770 + 3.37i)11-s + (0.0986 + 0.432i)13-s + (−1.57 + 0.104i)14-s + (1.24 − 1.56i)16-s + (1.92 + 0.925i)17-s + 3.99·19-s + (−1.41 − 0.682i)20-s + (−1.85 + 0.893i)22-s + (5.89 − 2.84i)23-s + ⋯
L(s)  = 1  + (0.0936 + 0.410i)2-s + (0.741 − 0.357i)4-s + (−0.266 − 0.334i)5-s + (−0.157 + 0.987i)7-s + (0.478 + 0.599i)8-s + (0.112 − 0.140i)10-s + (0.232 + 1.01i)11-s + (0.0273 + 0.119i)13-s + (−0.419 + 0.0279i)14-s + (0.311 − 0.391i)16-s + (0.466 + 0.224i)17-s + 0.915·19-s + (−0.316 − 0.152i)20-s + (−0.395 + 0.190i)22-s + (1.22 − 0.592i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.715 - 0.699i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.715 - 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60512 + 0.654310i\)
\(L(\frac12)\) \(\approx\) \(1.60512 + 0.654310i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.416 - 2.61i)T \)
good2 \( 1 + (-0.132 - 0.580i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (0.595 + 0.747i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (-0.770 - 3.37i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.0986 - 0.432i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-1.92 - 0.925i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 - 3.99T + 19T^{2} \)
23 \( 1 + (-5.89 + 2.84i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-2.93 - 1.41i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 8.81T + 31T^{2} \)
37 \( 1 + (-0.164 - 0.0792i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (6.84 + 8.58i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (5.25 - 6.58i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (0.430 + 1.88i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (6.76 - 3.25i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (5.19 - 6.51i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-1.37 - 0.664i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 - 6.48T + 67T^{2} \)
71 \( 1 + (3.77 - 1.82i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-2.13 + 9.37i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + 2.59T + 79T^{2} \)
83 \( 1 + (-1.73 + 7.60i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (1.43 - 6.28i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + 0.578T + 97T^{2} \)
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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

−11.33049285896428863014931074595, −10.33252605082232977012656836280, −9.386091254737249199200951301484, −8.458759740179390166917992933088, −7.38334514000295658528465243258, −6.61913577449659842565975986820, −5.53608720347868580430220982748, −4.75105530539110077997176696196, −3.03451045336671069851665385223, −1.71119896453997321799857115134, 1.25670813867114008910736116822, 3.19554300494449277532894339620, 3.57079705937972145141151237360, 5.20660491241677134096750150759, 6.57739800769103244801598110117, 7.27562612553937711383435420059, 8.053790622561369429001158805406, 9.394999188885169486263469519733, 10.36290782797273689142470807747, 11.24117585756931475001329958784

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