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Properties

Label 2-21e2-147.83-c1-0-2
Degree $2$
Conductor $441$
Sign $-0.397 - 0.917i$
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  + (−0.138 − 0.0315i)2-s + (−1.78 − 0.859i)4-s + (−0.192 + 0.241i)5-s + (0.709 + 2.54i)7-s + (0.441 + 0.352i)8-s + (0.0342 − 0.0272i)10-s + (−3.20 − 0.730i)11-s + (0.890 + 0.203i)13-s + (−0.0177 − 0.375i)14-s + (2.41 + 3.03i)16-s + (−6.36 + 3.06i)17-s − 1.38i·19-s + (0.549 − 0.264i)20-s + (0.419 + 0.202i)22-s + (−2.81 + 5.84i)23-s + ⋯
L(s)  = 1  + (−0.0978 − 0.0223i)2-s + (−0.891 − 0.429i)4-s + (−0.0859 + 0.107i)5-s + (0.268 + 0.963i)7-s + (0.156 + 0.124i)8-s + (0.0108 − 0.00862i)10-s + (−0.965 − 0.220i)11-s + (0.246 + 0.0563i)13-s + (−0.00473 − 0.100i)14-s + (0.604 + 0.758i)16-s + (−1.54 + 0.743i)17-s − 0.317i·19-s + (0.122 − 0.0592i)20-s + (0.0895 + 0.0431i)22-s + (−0.587 + 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.397 - 0.917i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.397 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.329144 + 0.501212i\)
\(L(\frac12)\) \(\approx\) \(0.329144 + 0.501212i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.709 - 2.54i)T \)
good2 \( 1 + (0.138 + 0.0315i)T + (1.80 + 0.867i)T^{2} \)
5 \( 1 + (0.192 - 0.241i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (3.20 + 0.730i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.890 - 0.203i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (6.36 - 3.06i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + 1.38iT - 19T^{2} \)
23 \( 1 + (2.81 - 5.84i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-1.58 - 3.29i)T + (-18.0 + 22.6i)T^{2} \)
31 \( 1 - 8.97iT - 31T^{2} \)
37 \( 1 + (-4.12 + 1.98i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (3.46 - 4.34i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (2.74 + 3.44i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-1.61 + 7.07i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-5.09 + 10.5i)T + (-33.0 - 41.4i)T^{2} \)
59 \( 1 + (5.11 + 6.41i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (1.81 + 3.76i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + (-2.21 + 4.59i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-7.82 + 1.78i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 - 3.00T + 79T^{2} \)
83 \( 1 + (-2.42 - 10.6i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-3.41 - 14.9i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + 1.26iT - 97T^{2} \)
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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

−11.19696506395387163987456731515, −10.57436879195458741674829076598, −9.461388466704225585172698692030, −8.714958143134460024836921522179, −8.081515071616482547273405230076, −6.65127170497088092718226236964, −5.50063485797629861593281890466, −4.87104503042242361771939409109, −3.46031343509282599978922486444, −1.87154727452038228790568781043, 0.39156268898619283943498560204, 2.61909371884823266526528006375, 4.29596821548852623481587073263, 4.57820211441644993822615468333, 6.13706455184546376847961772750, 7.41414793253099290319571498748, 8.079714367347595532458184018393, 8.934361564823593724945795790132, 10.00609387485974983454851325115, 10.66482058521210267226040192690

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