Properties

Label 2-21e2-441.142-c1-0-32
Degree $2$
Conductor $441$
Sign $0.984 + 0.177i$
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  + (−0.701 + 1.78i)2-s + (0.322 − 1.70i)3-s + (−1.23 − 1.14i)4-s + (0.740 − 0.356i)5-s + (2.81 + 1.76i)6-s + (0.756 − 2.53i)7-s + (−0.547 + 0.263i)8-s + (−2.79 − 1.09i)9-s + (0.117 + 1.57i)10-s + (1.56 + 1.96i)11-s + (−2.34 + 1.73i)12-s + (−0.227 + 0.579i)13-s + (3.99 + 3.12i)14-s + (−0.368 − 1.37i)15-s + (−0.338 − 4.52i)16-s + (2.66 − 2.47i)17-s + ⋯
L(s)  = 1  + (−0.495 + 1.26i)2-s + (0.186 − 0.982i)3-s + (−0.617 − 0.572i)4-s + (0.331 − 0.159i)5-s + (1.14 + 0.722i)6-s + (0.285 − 0.958i)7-s + (−0.193 + 0.0931i)8-s + (−0.930 − 0.366i)9-s + (0.0372 + 0.497i)10-s + (0.473 + 0.593i)11-s + (−0.677 + 0.499i)12-s + (−0.0630 + 0.160i)13-s + (1.06 + 0.836i)14-s + (−0.0950 − 0.355i)15-s + (−0.0846 − 1.13i)16-s + (0.646 − 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.984 + 0.177i$
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: \((2,\ 441,\ (\ :1/2),\ 0.984 + 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16449 - 0.103932i\)
\(L(\frac12)\) \(\approx\) \(1.16449 - 0.103932i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.322 + 1.70i)T \)
7 \( 1 + (-0.756 + 2.53i)T \)
good2 \( 1 + (0.701 - 1.78i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (-0.740 + 0.356i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-1.56 - 1.96i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.227 - 0.579i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-2.66 + 2.47i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-3.83 + 6.64i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.990 + 4.34i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (1.23 + 0.382i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-5.56 + 9.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.48 + 0.458i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-8.03 - 5.48i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (0.484 - 0.330i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (4.22 - 10.7i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-7.24 + 2.23i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (3.85 - 2.62i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (5.30 - 4.92i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (5.46 - 9.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.970 + 4.25i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.18 - 0.328i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-2.70 - 4.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.62 + 6.67i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (0.290 + 0.739i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (3.44 - 5.96i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.25966009239152781623461272704, −9.677648502898649130197259676083, −9.152817208875647207239264921226, −7.945173333742679297025420603533, −7.40890269987247251608632495572, −6.75252208865965542089509868141, −5.81290056160602861737244566596, −4.60461924967738891692449482194, −2.73421735264793237982764909812, −0.919944641701424301050506025535, 1.68338467562373143735302053333, 3.01879798476375013991547145545, 3.75432660227490131074150182615, 5.37523416430509323095572561875, 6.09809703754945038651710849792, 8.069496098996945170595973980904, 8.790127251502890525497916821313, 9.618169701764308979663639432877, 10.22067677723505321008590467042, 10.96932238758896227631228253113

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