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Properties

Label 2-444-37.11-c1-0-2
Degree $2$
Conductor $444$
Sign $0.989 + 0.146i$
Analytic cond. $3.54535$
Root an. cond. $1.88291$
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  + (−0.5 + 0.866i)3-s + (−3 − 1.73i)5-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + 6·11-s + (1.5 + 0.866i)13-s + (3 − 1.73i)15-s + (6 − 3.46i)17-s + (3 + 1.73i)19-s + (−0.499 − 0.866i)21-s − 3.46i·23-s + (3.5 + 6.06i)25-s + 0.999·27-s − 6.92i·29-s + 1.73i·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−1.34 − 0.774i)5-s + (−0.188 + 0.327i)7-s + (−0.166 − 0.288i)9-s + 1.80·11-s + (0.416 + 0.240i)13-s + (0.774 − 0.447i)15-s + (1.45 − 0.840i)17-s + (0.688 + 0.397i)19-s + (−0.109 − 0.188i)21-s − 0.722i·23-s + (0.700 + 1.21i)25-s + 0.192·27-s − 1.28i·29-s + 0.311i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.989 + 0.146i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ 0.989 + 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09558 - 0.0807272i\)
\(L(\frac12)\) \(\approx\) \(1.09558 - 0.0807272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (5 - 3.46i)T \)
good5 \( 1 + (3 + 1.73i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6 + 3.46i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
41 \( 1 + (6 - 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 12.1iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 - 1.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-9 + 5.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.73iT - 97T^{2} \)
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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

−11.44073481090929133702128067963, −10.09715224295158755465725677084, −9.208022976846388863639716569141, −8.516335226941541264038851481628, −7.49083261947874685092494477050, −6.36881781786042333648821517624, −5.19058648214797807514402718648, −4.13476642381958182566373018637, −3.43762632457338451517636771204, −0.976423742954294728741394703995, 1.19131287461873159632462798164, 3.39698377254026474132856635670, 3.89203993851820244067982668909, 5.58050717714227387758351222668, 6.75098917075190470870617041203, 7.28926154674775956289824025102, 8.190788651425422387238510747981, 9.282961808713625333709773574177, 10.51242375931511721908905377649, 11.24320851060650694253853204392

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