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Properties

Label 2-21e2-63.58-c1-0-33
Degree $2$
Conductor $441$
Sign $0.701 + 0.712i$
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  + 2.53·2-s + (0.592 − 1.62i)3-s + 4.41·4-s + (−0.439 − 0.761i)5-s + (1.50 − 4.12i)6-s + 6.10·8-s + (−2.29 − 1.92i)9-s + (−1.11 − 1.92i)10-s + (−1.93 + 3.35i)11-s + (2.61 − 7.18i)12-s + (−2.72 + 4.72i)13-s + (−1.50 + 0.264i)15-s + 6.63·16-s + (0.826 + 1.43i)17-s + (−5.81 − 4.88i)18-s + (1.20 − 2.08i)19-s + ⋯
L(s)  = 1  + 1.79·2-s + (0.342 − 0.939i)3-s + 2.20·4-s + (−0.196 − 0.340i)5-s + (0.612 − 1.68i)6-s + 2.15·8-s + (−0.766 − 0.642i)9-s + (−0.352 − 0.609i)10-s + (−0.584 + 1.01i)11-s + (0.754 − 2.07i)12-s + (−0.756 + 1.30i)13-s + (−0.387 + 0.0682i)15-s + 1.65·16-s + (0.200 + 0.347i)17-s + (−1.37 − 1.15i)18-s + (0.276 − 0.479i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.53856 - 1.48295i\)
\(L(\frac12)\) \(\approx\) \(3.53856 - 1.48295i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.592 + 1.62i)T \)
7 \( 1 \)
good2 \( 1 - 2.53T + 2T^{2} \)
5 \( 1 + (0.439 + 0.761i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.93 - 3.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.72 - 4.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.826 - 1.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.20 + 2.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.58 + 2.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.02 - 5.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.55T + 31T^{2} \)
37 \( 1 + (-2.27 + 3.94i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.592 - 1.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0923 + 0.160i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.02T + 47T^{2} \)
53 \( 1 + (3.64 + 6.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.66T + 59T^{2} \)
61 \( 1 - 2.59T + 61T^{2} \)
67 \( 1 + 2.95T + 67T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 + (6.39 + 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 5.95T + 79T^{2} \)
83 \( 1 + (0.109 + 0.189i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.51 + 9.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.25 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.68829049619080004196935970776, −10.43207467149162619054818324349, −9.087119401155107366364654254101, −7.82277933208719255010210178048, −6.97834343095587241663892024113, −6.32993511939749559004832764444, −5.01456496803002311951699913251, −4.31884687114070769975258971520, −2.87619731391955513877053745725, −1.94179657058725536028770949506, 2.81524444935790954324346269188, 3.20511444827342193602539892317, 4.41584685514155772557854079780, 5.36723169948720790610768949686, 5.95953337922919771716422695747, 7.44749340714635441326785572088, 8.242145537062966109060825866571, 9.815330816056983786568705291265, 10.60269139626965541680225281757, 11.36440303616484611332657064917

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