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Properties

Label 2-21e2-9.2-c2-0-54
Degree $2$
Conductor $441$
Sign $-0.894 - 0.447i$
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  + (−2.37 − 1.37i)2-s + (2.78 − 1.10i)3-s + (1.76 + 3.06i)4-s + (−2.32 + 1.34i)5-s + (−8.14 − 1.18i)6-s + 1.27i·8-s + (6.53 − 6.18i)9-s + 7.37·10-s + (−7.18 − 4.14i)11-s + (8.32 + 6.57i)12-s + (1.91 + 3.31i)13-s + (−4.99 + 6.32i)15-s + (8.82 − 15.2i)16-s − 16.8i·17-s + (−24.0 + 5.73i)18-s − 9.54·19-s + ⋯
L(s)  = 1  + (−1.18 − 0.686i)2-s + (0.929 − 0.369i)3-s + (0.441 + 0.765i)4-s + (−0.465 + 0.268i)5-s + (−1.35 − 0.197i)6-s + 0.159i·8-s + (0.726 − 0.687i)9-s + 0.737·10-s + (−0.653 − 0.377i)11-s + (0.693 + 0.547i)12-s + (0.147 + 0.255i)13-s + (−0.332 + 0.421i)15-s + (0.551 − 0.954i)16-s − 0.993i·17-s + (−1.33 + 0.318i)18-s − 0.502·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ -0.894 - 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0849704 + 0.359732i\)
\(L(\frac12)\) \(\approx\) \(0.0849704 + 0.359732i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.78 + 1.10i)T \)
7 \( 1 \)
good2 \( 1 + (2.37 + 1.37i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (2.32 - 1.34i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (7.18 + 4.14i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.91 - 3.31i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 16.8iT - 289T^{2} \)
19 \( 1 + 9.54T + 361T^{2} \)
23 \( 1 + (21.1 - 12.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (41.1 + 23.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (19.6 + 33.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 46.6T + 1.36e3T^{2} \)
41 \( 1 + (51.2 - 29.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-28.0 + 48.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (18.6 + 10.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 25.6iT - 2.80e3T^{2} \)
59 \( 1 + (40.1 - 23.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (30.1 - 52.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-32.0 - 55.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 49.6iT - 5.04e3T^{2} \)
73 \( 1 + 117.T + 5.32e3T^{2} \)
79 \( 1 + (20.2 - 35.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (64.7 + 37.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 69.2iT - 7.92e3T^{2} \)
97 \( 1 + (46.9 - 81.3i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.12729361494691039001021904127, −9.493084977232850497874802966824, −8.678810062408868547499346258254, −7.78769242606292771734467728744, −7.33428022338668715620517932434, −5.74550863083521439432024236388, −4.00389518662516753484055277204, −2.83440729951119438354144098383, −1.83104542501761608913831086084, −0.19670121821897849674900999754, 1.81260871833609993223277356948, 3.50307841922487776241627815324, 4.54865550607639373984479139739, 6.07802223319684042175044450169, 7.31494083128697698287602284218, 8.006791559196470819257702216840, 8.537146432192320028632759007378, 9.391571659386476180826888605579, 10.25680192866760298900286715669, 10.86339379074746413613919138882

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