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Properties

Label 2-21e2-63.11-c2-0-60
Degree $2$
Conductor $441$
Sign $-0.963 + 0.267i$
Analytic cond. $12.0163$
Root an. cond. $3.46646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.22i·2-s + (2.32 − 1.89i)3-s − 6.39·4-s + (4.79 + 2.76i)5-s + (−6.10 − 7.50i)6-s + 7.70i·8-s + (1.82 − 8.81i)9-s + (8.91 − 15.4i)10-s + (15.3 − 8.84i)11-s + (−14.8 + 12.1i)12-s + (−2.03 − 3.52i)13-s + (16.3 − 2.63i)15-s − 0.715·16-s + (14.3 + 8.27i)17-s + (−28.4 − 5.89i)18-s + (3.92 + 6.79i)19-s + ⋯
L(s)  = 1  − 1.61i·2-s + (0.775 − 0.631i)3-s − 1.59·4-s + (0.958 + 0.553i)5-s + (−1.01 − 1.25i)6-s + 0.963i·8-s + (0.203 − 0.979i)9-s + (0.891 − 1.54i)10-s + (1.39 − 0.804i)11-s + (−1.23 + 1.00i)12-s + (−0.156 − 0.271i)13-s + (1.09 − 0.175i)15-s − 0.0447·16-s + (0.843 + 0.486i)17-s + (−1.57 − 0.327i)18-s + (0.206 + 0.357i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.963 + 0.267i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ -0.963 + 0.267i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.350940 - 2.57191i\)
\(L(\frac12)\) \(\approx\) \(0.350940 - 2.57191i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.32 + 1.89i)T \)
7 \( 1 \)
good2 \( 1 + 3.22iT - 4T^{2} \)
5 \( 1 + (-4.79 - 2.76i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-15.3 + 8.84i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (2.03 + 3.52i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-14.3 - 8.27i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-3.92 - 6.79i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-8.71 - 5.03i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (39.9 + 23.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 29.6T + 961T^{2} \)
37 \( 1 + (-15.5 - 27.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (27.8 - 16.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-3.35 + 5.80i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 16.4iT - 2.20e3T^{2} \)
53 \( 1 + (32.5 + 18.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 95.0iT - 3.48e3T^{2} \)
61 \( 1 - 73.7T + 3.72e3T^{2} \)
67 \( 1 + 12.1T + 4.48e3T^{2} \)
71 \( 1 + 20.0iT - 5.04e3T^{2} \)
73 \( 1 + (-11.4 + 19.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 138.T + 6.24e3T^{2} \)
83 \( 1 + (13.6 + 7.90i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-46.9 + 27.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (86.1 - 149. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.49830582589722565335630226631, −9.601379681590491570835907700886, −9.180044392071638119138302391176, −8.019687020888444294476171453214, −6.71262293091207148003803030109, −5.76741399593863844214523823864, −3.86334168441467159636778515213, −3.18489492856143259873800933809, −2.03569175784583446557566211447, −1.13683143854418589923234879071, 1.85885814727344892628267879938, 3.77905020102680760872642753795, 4.91663310159236630798322264194, 5.55542014446646842189064185420, 6.82384130871225062088286571589, 7.51736085943009915732546787948, 8.688052779691573140598940253603, 9.429731919627417288276685204727, 9.604708641663550953854768186364, 11.17023969760776307339678811389

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