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Properties

Label 2-21e2-147.101-c1-0-14
Degree $2$
Conductor $441$
Sign $0.628 - 0.777i$
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  + (2.21 + 0.870i)2-s + (2.69 + 2.50i)4-s + (1.55 + 1.06i)5-s + (0.00333 − 2.64i)7-s + (1.73 + 3.60i)8-s + (2.53 + 3.71i)10-s + (0.0972 + 0.645i)11-s + (0.531 − 0.424i)13-s + (2.31 − 5.86i)14-s + (0.163 + 2.18i)16-s + (−5.58 − 1.72i)17-s + (−5.58 + 3.22i)19-s + (1.54 + 6.76i)20-s + (−0.346 + 1.51i)22-s + (2.20 + 7.15i)23-s + ⋯
L(s)  = 1  + (1.56 + 0.615i)2-s + (1.34 + 1.25i)4-s + (0.696 + 0.474i)5-s + (0.00125 − 0.999i)7-s + (0.614 + 1.27i)8-s + (0.800 + 1.17i)10-s + (0.0293 + 0.194i)11-s + (0.147 − 0.117i)13-s + (0.617 − 1.56i)14-s + (0.0408 + 0.545i)16-s + (−1.35 − 0.417i)17-s + (−1.28 + 0.739i)19-s + (0.345 + 1.51i)20-s + (−0.0737 + 0.323i)22-s + (0.460 + 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.628 - 0.777i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.628 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.07970 + 1.47021i\)
\(L(\frac12)\) \(\approx\) \(3.07970 + 1.47021i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.00333 + 2.64i)T \)
good2 \( 1 + (-2.21 - 0.870i)T + (1.46 + 1.36i)T^{2} \)
5 \( 1 + (-1.55 - 1.06i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.0972 - 0.645i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-0.531 + 0.424i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (5.58 + 1.72i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (5.58 - 3.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.20 - 7.15i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-4.23 + 0.966i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (2.43 + 1.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.42 - 2.25i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (-3.48 + 1.67i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (2.03 + 0.982i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.26 + 3.22i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-6.96 + 7.50i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (1.50 - 1.02i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (6.85 + 7.39i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (5.80 - 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-15.2 - 3.47i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-8.21 + 3.22i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (-4.60 - 7.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.98 - 7.50i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (4.81 + 0.725i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + 8.08iT - 97T^{2} \)
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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

−11.37277272457875200822537184110, −10.58906813226433772414384838255, −9.594703870325363974947418300539, −8.152618448127200650579052818146, −6.99473692210815468229477407250, −6.55503401461976899040636767726, −5.53302776827274281548917024538, −4.45204485329734308330541102261, −3.61597609453441881043541317523, −2.23711238285490496546629259342, 1.95324448048581853185673426347, 2.76845210527343777903772239802, 4.30376230873650264365394355561, 5.01090145910447670146842791552, 6.05643892068602598233346071397, 6.60971805222273629035904245974, 8.597684197346431220311796992575, 9.117504430572554658806492461056, 10.61378159793145420540994189214, 11.09103454159140914708759085197

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