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Properties

Label 2-21e2-63.25-c1-0-28
Degree $2$
Conductor $441$
Sign $0.989 - 0.146i$
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  + 2.05·2-s + (1.70 + 0.283i)3-s + 2.21·4-s + (−0.0731 + 0.126i)5-s + (3.50 + 0.582i)6-s + 0.446·8-s + (2.83 + 0.969i)9-s + (−0.150 + 0.260i)10-s + (−0.832 − 1.44i)11-s + (3.78 + 0.628i)12-s + (−0.0999 − 0.173i)13-s + (−0.160 + 0.195i)15-s − 3.51·16-s + (−3.13 + 5.43i)17-s + (5.83 + 1.99i)18-s + (−3.45 − 5.99i)19-s + ⋯
L(s)  = 1  + 1.45·2-s + (0.986 + 0.163i)3-s + 1.10·4-s + (−0.0327 + 0.0566i)5-s + (1.43 + 0.237i)6-s + 0.157·8-s + (0.946 + 0.323i)9-s + (−0.0474 + 0.0822i)10-s + (−0.250 − 0.434i)11-s + (1.09 + 0.181i)12-s + (−0.0277 − 0.0480i)13-s + (−0.0415 + 0.0505i)15-s − 0.879·16-s + (−0.760 + 1.31i)17-s + (1.37 + 0.469i)18-s + (−0.793 − 1.37i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.989 - 0.146i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.989 - 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.64735 + 0.268303i\)
\(L(\frac12)\) \(\approx\) \(3.64735 + 0.268303i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 - 0.283i)T \)
7 \( 1 \)
good2 \( 1 - 2.05T + 2T^{2} \)
5 \( 1 + (0.0731 - 0.126i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.832 + 1.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0999 + 0.173i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.13 - 5.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.45 + 5.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.09 + 5.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.46 - 4.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.51T + 31T^{2} \)
37 \( 1 + (3.50 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.15 + 2.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.940 - 1.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.81T + 47T^{2} \)
53 \( 1 + (2.67 - 4.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.57T + 59T^{2} \)
61 \( 1 - 0.678T + 61T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 + (-0.778 + 1.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (3.75 - 6.50i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.53 + 7.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.98 + 6.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.10118797415342173590516500736, −10.60117850138376024193883960700, −9.033663727860207314155962806984, −8.610206421645845529672189922353, −7.18213173188078507513822362249, −6.37457125916462719826121084635, −5.06613204638275187404922392982, −4.23595147968851732016820678304, −3.25453694760961284341379274111, −2.26319804427224440741408921516, 2.12273432553517660731884850959, 3.17312869262596026090421207065, 4.20752881099597758829632750848, 5.04508541911414919321160973490, 6.35489732423258101936188362701, 7.20269110808633851380625766160, 8.275612142936481435518437751810, 9.276881674252171908611991934834, 10.19079742436438029898394526330, 11.50352029200091292872760661673

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