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Properties

Label 2-21e2-7.3-c2-0-18
Degree $2$
Conductor $441$
Sign $0.832 + 0.553i$
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  + (0.5 + 0.866i)2-s + (1.5 − 2.59i)4-s + (−6 + 3.46i)5-s + 7·8-s + (−6 − 3.46i)10-s + (5 − 8.66i)11-s − 6.92i·13-s + (−2.5 − 4.33i)16-s + (18 − 10.3i)19-s + 20.7i·20-s + 10·22-s + (−7 − 12.1i)23-s + (11.5 − 19.9i)25-s + (5.99 − 3.46i)26-s + 38·29-s + ⋯
L(s)  = 1  + (0.250 + 0.433i)2-s + (0.375 − 0.649i)4-s + (−1.20 + 0.692i)5-s + 0.875·8-s + (−0.600 − 0.346i)10-s + (0.454 − 0.787i)11-s − 0.532i·13-s + (−0.156 − 0.270i)16-s + (0.947 − 0.546i)19-s + 1.03i·20-s + 0.454·22-s + (−0.304 − 0.527i)23-s + (0.460 − 0.796i)25-s + (0.230 − 0.133i)26-s + 1.31·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.68119 - 0.508081i\)
\(L(\frac12)\) \(\approx\) \(1.68119 - 0.508081i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + (6 - 3.46i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-5 + 8.66i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 6.92iT - 169T^{2} \)
17 \( 1 + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-18 + 10.3i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (7 + 12.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 38T + 841T^{2} \)
31 \( 1 + (-24 - 13.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (13 + 22.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 69.2iT - 1.68e3T^{2} \)
43 \( 1 - 26T + 1.84e3T^{2} \)
47 \( 1 + (-24 + 13.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (66 + 38.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (30 - 17.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (37 - 64.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 62T + 5.04e3T^{2} \)
73 \( 1 + (36 + 20.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-23 - 39.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 90.0iT - 6.88e3T^{2} \)
89 \( 1 + (36 - 20.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 55.4iT - 9.40e3T^{2} \)
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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

−10.85232218184138291639017060738, −10.22066068564005403628841892910, −8.862154560786352882588307162043, −7.81821508609394004378138709673, −7.06523977111531570455009473303, −6.23762758513873042972665238740, −5.14335909210808531857777258859, −3.93353560292795044190753287446, −2.78440997904297778289060084613, −0.76344746721236627343897986072, 1.40211559130094202822623452728, 3.03282956733126414202790227349, 4.13968391053081715446666400413, 4.72186823663490698432441608196, 6.45796781526808669448142055201, 7.56885607097105294795856732649, 8.036844741721526858985531984178, 9.143940278279917636026403565259, 10.21092949627549408797665980272, 11.40796252848335363002229291088

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