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Properties

Label 2-21e2-9.4-c1-0-0
Degree $2$
Conductor $441$
Sign $-0.958 - 0.285i$
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  + (−0.649 − 1.12i)2-s + (0.0514 + 1.73i)3-s + (0.155 − 0.268i)4-s + (−1.76 + 3.05i)5-s + (1.91 − 1.18i)6-s − 3.00·8-s + (−2.99 + 0.177i)9-s + 4.58·10-s + (−0.589 − 1.02i)11-s + (0.473 + 0.254i)12-s + (1.61 − 2.78i)13-s + (−5.37 − 2.89i)15-s + (1.64 + 2.84i)16-s − 4.90·17-s + (2.14 + 3.25i)18-s − 6.86·19-s + ⋯
L(s)  = 1  + (−0.459 − 0.796i)2-s + (0.0296 + 0.999i)3-s + (0.0775 − 0.134i)4-s + (−0.788 + 1.36i)5-s + (0.782 − 0.482i)6-s − 1.06·8-s + (−0.998 + 0.0593i)9-s + 1.44·10-s + (−0.177 − 0.307i)11-s + (0.136 + 0.0735i)12-s + (0.446 − 0.773i)13-s + (−1.38 − 0.747i)15-s + (0.410 + 0.710i)16-s − 1.18·17-s + (0.505 + 0.767i)18-s − 1.57·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.958 - 0.285i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.958 - 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0225708 + 0.154730i\)
\(L(\frac12)\) \(\approx\) \(0.0225708 + 0.154730i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0514 - 1.73i)T \)
7 \( 1 \)
good2 \( 1 + (0.649 + 1.12i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.76 - 3.05i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.589 + 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.61 + 2.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.90T + 17T^{2} \)
19 \( 1 + 6.86T + 19T^{2} \)
23 \( 1 + (-2.14 + 3.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.36 - 2.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.960 + 1.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.76T + 37T^{2} \)
41 \( 1 + (3.32 - 5.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.83 - 8.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.316 + 0.548i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.22T + 53T^{2} \)
59 \( 1 + (4.10 - 7.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.82 - 8.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.66 - 4.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 - 1.03T + 73T^{2} \)
79 \( 1 + (0.502 + 0.869i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.65 + 6.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (5.46 + 9.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.05048750857103924016966953363, −10.69133161918871877537627450072, −10.22401509944018461348064609311, −8.922008224307776335114878645756, −8.288169706053249724277016760422, −6.77461534236130558030998965565, −5.98609221707394469173196362372, −4.43055115749419655590934976866, −3.28372909676779934955594930059, −2.55345090321154107067253580079, 0.10506758023334002040720403292, 1.99152303730021979734083540786, 3.82693691047203671505122142552, 5.10155348483201815439581248987, 6.39007475094477538203471948923, 7.07474484025545943427596558977, 8.029278138075768031090940737078, 8.724668617025452115319964106489, 9.057045537475360588256993653656, 10.95990675390621397568489477854

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