Properties

Label 2-21e2-441.142-c1-0-41
Degree $2$
Conductor $441$
Sign $-0.434 + 0.900i$
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  + (−0.0656 + 0.167i)2-s + (−1.14 − 1.29i)3-s + (1.44 + 1.33i)4-s + (−0.436 + 0.209i)5-s + (0.292 − 0.106i)6-s + (−2.25 − 1.38i)7-s + (−0.641 + 0.309i)8-s + (−0.369 + 2.97i)9-s + (−0.00649 − 0.0866i)10-s + (−3.81 − 4.78i)11-s + (0.0827 − 3.40i)12-s + (1.89 − 4.82i)13-s + (0.379 − 0.285i)14-s + (0.772 + 0.325i)15-s + (0.284 + 3.79i)16-s + (0.806 − 0.747i)17-s + ⋯
L(s)  = 1  + (−0.0463 + 0.118i)2-s + (−0.662 − 0.749i)3-s + (0.721 + 0.669i)4-s + (−0.195 + 0.0939i)5-s + (0.119 − 0.0435i)6-s + (−0.851 − 0.523i)7-s + (−0.226 + 0.109i)8-s + (−0.123 + 0.992i)9-s + (−0.00205 − 0.0274i)10-s + (−1.14 − 1.44i)11-s + (0.0239 − 0.983i)12-s + (0.525 − 1.33i)13-s + (0.101 − 0.0763i)14-s + (0.199 + 0.0839i)15-s + (0.0711 + 0.949i)16-s + (0.195 − 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.434 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.390314 - 0.621602i\)
\(L(\frac12)\) \(\approx\) \(0.390314 - 0.621602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.14 + 1.29i)T \)
7 \( 1 + (2.25 + 1.38i)T \)
good2 \( 1 + (0.0656 - 0.167i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (0.436 - 0.209i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (3.81 + 4.78i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.89 + 4.82i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-0.806 + 0.747i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-1.79 + 3.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.737 + 3.23i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (7.60 + 2.34i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-1.23 + 2.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.90 + 0.894i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (4.21 + 2.87i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-9.04 + 6.16i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (1.72 - 4.40i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-1.23 + 0.379i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-1.67 + 1.13i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (6.03 - 5.60i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (2.87 - 4.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.22 - 5.36i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-8.43 - 1.27i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-3.21 - 5.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.79 + 14.7i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-2.89 - 7.37i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-0.729 + 1.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.94419470488443668345728622181, −10.36828917724210960881101801085, −8.710991305660334229087228845361, −7.72654256018056714280350193718, −7.30614835762608686708397053138, −6.09508179104382627795674456913, −5.53118039356913949024643546036, −3.53801278029990585711202268388, −2.67977661217591470477680614221, −0.47488485735718897778287396288, 1.94515355615638882086928719943, 3.47840537190507997631886308603, 4.80246522677885951129586921258, 5.74514159089965364482778940574, 6.54831499986148645073911373855, 7.52966957005228698067067696598, 9.161178134229130917164882496116, 9.795223281552736532115096874314, 10.40524097652799908213091905188, 11.38459269269534202980436247594

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