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Properties

Label 2-21e2-63.38-c1-0-11
Degree $2$
Conductor $441$
Sign $-0.0940 + 0.995i$
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  − 2.57i·2-s + (−1.08 + 1.34i)3-s − 4.64·4-s + (−1.16 + 2.01i)5-s + (3.47 + 2.80i)6-s + 6.82i·8-s + (−0.639 − 2.93i)9-s + (5.20 + 3.00i)10-s + (3.78 − 2.18i)11-s + (5.04 − 6.26i)12-s + (1.14 − 0.660i)13-s + (−1.45 − 3.76i)15-s + 8.30·16-s + (2.89 − 5.01i)17-s + (−7.55 + 1.64i)18-s + (−0.584 + 0.337i)19-s + ⋯
L(s)  = 1  − 1.82i·2-s + (−0.627 + 0.778i)3-s − 2.32·4-s + (−0.521 + 0.903i)5-s + (1.41 + 1.14i)6-s + 2.41i·8-s + (−0.213 − 0.976i)9-s + (1.64 + 0.950i)10-s + (1.14 − 0.658i)11-s + (1.45 − 1.80i)12-s + (0.317 − 0.183i)13-s + (−0.376 − 0.972i)15-s + 2.07·16-s + (0.701 − 1.21i)17-s + (−1.78 + 0.388i)18-s + (−0.134 + 0.0774i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0940 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0940 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0940 + 0.995i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.0940 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.619086 - 0.680328i\)
\(L(\frac12)\) \(\approx\) \(0.619086 - 0.680328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08 - 1.34i)T \)
7 \( 1 \)
good2 \( 1 + 2.57iT - 2T^{2} \)
5 \( 1 + (1.16 - 2.01i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.78 + 2.18i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.14 + 0.660i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.89 + 5.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.584 - 0.337i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.81 - 2.78i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.86 - 2.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.01iT - 31T^{2} \)
37 \( 1 + (1.50 + 2.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.29 - 5.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.89 + 6.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.493T + 47T^{2} \)
53 \( 1 + (-3.59 - 2.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.31T + 59T^{2} \)
61 \( 1 - 2.05iT - 61T^{2} \)
67 \( 1 + 4.82T + 67T^{2} \)
71 \( 1 + 1.17iT - 71T^{2} \)
73 \( 1 + (13.0 + 7.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + (-5.32 + 9.22i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.66 - 2.87i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.7 - 7.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.98264899768360081611604050257, −10.35856998075109479260934353867, −9.409688834157962340455540878166, −8.804577660327106684004460884394, −7.17939435095613477966316342399, −5.80434356843538595282883327048, −4.59472744446475873937017147073, −3.57437747994555021954034252704, −3.01424825794412710267444298058, −0.894668892673546882644882185066, 1.08663765604848366293023896886, 4.15000271460021773924017594598, 4.90614924478379399042337034934, 5.98239811396253240398189744295, 6.69049196076226895899633728039, 7.50449139648044896306599104964, 8.437194592588167872220810934138, 8.912592273408715143867289616745, 10.26247606096440199582533402789, 11.68035383255368929596148920298

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