Properties

Label 2-21e2-63.25-c1-0-26
Degree $2$
Conductor $441$
Sign $-0.843 + 0.537i$
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  − 2.46·2-s + (0.796 − 1.53i)3-s + 4.05·4-s + (1.29 − 2.24i)5-s + (−1.96 + 3.78i)6-s − 5.05·8-s + (−1.73 − 2.45i)9-s + (−3.19 + 5.52i)10-s + (−2.25 − 3.90i)11-s + (3.23 − 6.23i)12-s + (−0.5 − 0.866i)13-s + (−2.42 − 3.78i)15-s + 4.32·16-s + (0.472 − 0.819i)17-s + (4.25 + 6.03i)18-s + (2.02 + 3.51i)19-s + ⋯
L(s)  = 1  − 1.73·2-s + (0.460 − 0.887i)3-s + 2.02·4-s + (0.579 − 1.00i)5-s + (−0.800 + 1.54i)6-s − 1.78·8-s + (−0.576 − 0.816i)9-s + (−1.00 + 1.74i)10-s + (−0.680 − 1.17i)11-s + (0.932 − 1.79i)12-s + (−0.138 − 0.240i)13-s + (−0.625 − 0.977i)15-s + 1.08·16-s + (0.114 − 0.198i)17-s + (1.00 + 1.42i)18-s + (0.465 + 0.805i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.843 + 0.537i$
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: \((2,\ 441,\ (\ :1/2),\ -0.843 + 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183565 - 0.630099i\)
\(L(\frac12)\) \(\approx\) \(0.183565 - 0.630099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.796 + 1.53i)T \)
7 \( 1 \)
good2 \( 1 + 2.46T + 2T^{2} \)
5 \( 1 + (-1.29 + 2.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.25 + 3.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.472 + 0.819i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.02 - 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.136 + 0.236i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.32T + 31T^{2} \)
37 \( 1 + (0.890 + 1.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.20 - 5.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.21 + 9.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + (-3.13 + 5.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.72T + 59T^{2} \)
61 \( 1 + 2.27T + 61T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 + (-0.753 + 1.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + (-0.472 + 0.819i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.17 - 12.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.74 + 9.95i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.48821605625654465049055369525, −9.524665732716706008213502545705, −8.858598952517301195744063517664, −8.171576496201363675115677840350, −7.55668068040645026476143702405, −6.34782484963807183365242019088, −5.42032082552388186942971863224, −3.08045787666799646134911828145, −1.76992940563860351208040119292, −0.67161604404873578446540322970, 2.10451806765471578826005338199, 2.92582215667111395665157219090, 4.72888735364825148093065206085, 6.22832448637305968611501454102, 7.28938682788753489207558936314, 7.929570557177715228705384339040, 9.127214916862675779053661560559, 9.667226703617786667081042533752, 10.35501538622504197566492048049, 10.84835475041183943836660195026

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