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Properties

Label 2-21e2-63.59-c1-0-32
Degree $2$
Conductor $441$
Sign $0.174 - 0.984i$
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  + (−2.23 − 1.28i)2-s + (0.625 − 1.61i)3-s + (2.32 + 4.02i)4-s − 2.33·5-s + (−3.47 + 2.80i)6-s − 6.82i·8-s + (−2.21 − 2.01i)9-s + (5.20 + 3.00i)10-s − 4.36i·11-s + (7.95 − 1.23i)12-s + (−1.14 − 0.660i)13-s + (−1.45 + 3.76i)15-s + (−4.15 + 7.18i)16-s + (−2.89 + 5.01i)17-s + (2.34 + 7.36i)18-s + (−0.584 + 0.337i)19-s + ⋯
L(s)  = 1  + (−1.57 − 0.911i)2-s + (0.360 − 0.932i)3-s + (1.16 + 2.01i)4-s − 1.04·5-s + (−1.41 + 1.14i)6-s − 2.41i·8-s + (−0.739 − 0.673i)9-s + (1.64 + 0.950i)10-s − 1.31i·11-s + (2.29 − 0.357i)12-s + (−0.317 − 0.183i)13-s + (−0.376 + 0.972i)15-s + (−1.03 + 1.79i)16-s + (−0.701 + 1.21i)17-s + (0.553 + 1.73i)18-s + (−0.134 + 0.0774i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.174 - 0.984i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.174 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0302539 + 0.0253753i\)
\(L(\frac12)\) \(\approx\) \(0.0302539 + 0.0253753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.625 + 1.61i)T \)
7 \( 1 \)
good2 \( 1 + (2.23 + 1.28i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2.33T + 5T^{2} \)
11 \( 1 + 4.36iT - 11T^{2} \)
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 \( 1 - 5.56iT - 23T^{2} \)
29 \( 1 + (-3.86 + 2.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.47 - 2.00i)T + (15.5 - 26.8i)T^{2} \)
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 \( 1 + (-0.246 + 0.427i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.59 + 2.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.15 - 3.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.77 + 1.02i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.41 - 4.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.17iT - 71T^{2} \)
73 \( 1 + (13.0 + 7.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.30 + 9.18i)T + (-39.5 - 68.4i)T^{2} \)
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

−10.55355686022495419765975787698, −9.315585588254219382381152561635, −8.477109837145483560281770243681, −8.039808397177956236735678865279, −7.26902084516652372545819389579, −6.11687457674881939125910184629, −3.77454703647699951261769554770, −2.90819953388083602682799057131, −1.48247510127311906140192498267, −0.03971674332999279436738967696, 2.37988849342582976067569937786, 4.24313056232778467415735414292, 5.18640796200845907618037680849, 6.78726059752830981832108989124, 7.40690122681167999653981689904, 8.314611946485303159857985465562, 9.041947246745157640701493297520, 9.772146584770367029923768846034, 10.54433183163069317677519309559, 11.34714536331430420593593318785

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