Properties

Label 2-21e2-441.4-c1-0-0
Degree $2$
Conductor $441$
Sign $0.480 + 0.876i$
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  + (−0.173 + 2.31i)2-s + (−0.732 + 1.56i)3-s + (−3.32 − 0.501i)4-s + (−0.445 + 1.95i)5-s + (−3.49 − 1.96i)6-s + (0.693 − 2.55i)7-s + (0.704 − 3.08i)8-s + (−1.92 − 2.29i)9-s + (−4.42 − 1.36i)10-s + (−2.98 + 1.43i)11-s + (3.22 − 4.85i)12-s + (0.277 − 3.69i)13-s + (5.77 + 2.04i)14-s + (−2.73 − 2.12i)15-s + (0.572 + 0.176i)16-s + (−5.17 + 0.779i)17-s + ⋯
L(s)  = 1  + (−0.122 + 1.63i)2-s + (−0.422 + 0.906i)3-s + (−1.66 − 0.250i)4-s + (−0.199 + 0.872i)5-s + (−1.42 − 0.801i)6-s + (0.262 − 0.965i)7-s + (0.248 − 1.09i)8-s + (−0.642 − 0.766i)9-s + (−1.40 − 0.431i)10-s + (−0.901 + 0.434i)11-s + (0.931 − 1.40i)12-s + (0.0768 − 1.02i)13-s + (1.54 + 0.546i)14-s + (−0.706 − 0.549i)15-s + (0.143 + 0.0441i)16-s + (−1.25 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.480 + 0.876i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.480 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.322579 - 0.191022i\)
\(L(\frac12)\) \(\approx\) \(0.322579 - 0.191022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.732 - 1.56i)T \)
7 \( 1 + (-0.693 + 2.55i)T \)
good2 \( 1 + (0.173 - 2.31i)T + (-1.97 - 0.298i)T^{2} \)
5 \( 1 + (0.445 - 1.95i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (2.98 - 1.43i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.277 + 3.69i)T + (-12.8 - 1.93i)T^{2} \)
17 \( 1 + (5.17 - 0.779i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-0.930 - 1.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.48 - 1.86i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (1.83 - 4.68i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + (-0.858 - 1.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.33 - 8.50i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (0.0655 + 0.0607i)T + (3.06 + 40.8i)T^{2} \)
43 \( 1 + (3.16 - 2.93i)T + (3.21 - 42.8i)T^{2} \)
47 \( 1 + (-0.723 + 9.65i)T + (-46.4 - 7.00i)T^{2} \)
53 \( 1 + (-0.956 - 2.43i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (9.73 - 9.03i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (6.63 - 0.999i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (-3.55 - 6.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.33 + 7.94i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (6.80 - 4.64i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-5.12 + 8.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.118 + 1.58i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (1.11 + 14.9i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (0.0150 + 0.0261i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.57741773332685587610177798664, −10.53898150392592823531437657073, −10.25362287310533291743912702992, −8.938493194777001251079114537568, −7.972944829031212350570138076084, −7.18548650995418564150130391667, −6.39065450789999909849576988034, −5.30518049887348035781812870085, −4.54323233560797097583481296691, −3.26818691778818847149337961443, 0.24970434819922884343897341012, 1.84310229227244964540963349847, 2.63698965296286405582510363808, 4.38578676591829868526014344042, 5.26290803714060558873136485211, 6.51664784270130398278916479934, 8.012878195811705969549322560711, 8.789583473537085817127221119906, 9.427842552190758534155290710421, 10.94541152944507961923595744778

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