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Properties

Label 2-444-111.2-c1-0-12
Degree $2$
Conductor $444$
Sign $-0.808 + 0.589i$
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  + (1.21 + 1.23i)3-s + (−3.88 − 1.80i)5-s + (−2.30 + 0.838i)7-s + (−0.0472 + 2.99i)9-s + (−1.76 − 3.05i)11-s + (−1.26 − 1.80i)13-s + (−2.48 − 6.98i)15-s + (4.04 − 5.77i)17-s + (−4.28 + 0.375i)19-s + (−3.83 − 1.82i)21-s + (−8.20 + 2.19i)23-s + (8.57 + 10.2i)25-s + (−3.76 + 3.58i)27-s + (−5.54 − 1.48i)29-s + (2.89 + 2.89i)31-s + ⋯
L(s)  = 1  + (0.701 + 0.712i)3-s + (−1.73 − 0.809i)5-s + (−0.871 + 0.317i)7-s + (−0.0157 + 0.999i)9-s + (−0.532 − 0.922i)11-s + (−0.351 − 0.501i)13-s + (−0.640 − 1.80i)15-s + (0.980 − 1.40i)17-s + (−0.983 + 0.0860i)19-s + (−0.837 − 0.398i)21-s + (−1.71 + 0.458i)23-s + (1.71 + 2.04i)25-s + (−0.723 + 0.690i)27-s + (−1.02 − 0.275i)29-s + (0.520 + 0.520i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 + 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.808 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $-0.808 + 0.589i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ -0.808 + 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0697800 - 0.214132i\)
\(L(\frac12)\) \(\approx\) \(0.0697800 - 0.214132i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.21 - 1.23i)T \)
37 \( 1 + (-3.85 + 4.70i)T \)
good5 \( 1 + (3.88 + 1.80i)T + (3.21 + 3.83i)T^{2} \)
7 \( 1 + (2.30 - 0.838i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (1.76 + 3.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.26 + 1.80i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-4.04 + 5.77i)T + (-5.81 - 15.9i)T^{2} \)
19 \( 1 + (4.28 - 0.375i)T + (18.7 - 3.29i)T^{2} \)
23 \( 1 + (8.20 - 2.19i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (5.54 + 1.48i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-2.89 - 2.89i)T + 31iT^{2} \)
41 \( 1 + (0.364 - 2.06i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.97 - 1.97i)T - 43iT^{2} \)
47 \( 1 + (-3.93 - 2.27i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.26 - 6.21i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.868 + 1.86i)T + (-37.9 + 45.1i)T^{2} \)
61 \( 1 + (3.12 - 2.18i)T + (20.8 - 57.3i)T^{2} \)
67 \( 1 + (-0.274 - 0.755i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-3.02 + 3.60i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + 7.16iT - 73T^{2} \)
79 \( 1 + (-4.42 + 9.48i)T + (-50.7 - 60.5i)T^{2} \)
83 \( 1 + (3.53 - 0.623i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-5.25 + 2.45i)T + (57.2 - 68.1i)T^{2} \)
97 \( 1 + (3.78 + 14.1i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.74294075611157141755640320647, −9.683018105299137964643703138939, −8.914278414340663293731780217453, −7.951657218524542004520895059220, −7.61571557040474178934954710962, −5.77721213723874389363684880243, −4.70137125368315594752131876549, −3.69441497356200421095878744662, −2.92781202700336022140015418060, −0.12274929129991573731117123687, 2.30823962439806722668449354080, 3.60182655245013930705501428217, 4.15359614876493415164721393112, 6.30307495895577869555153261373, 7.00555602511917395348995134303, 7.85241540726651290182120125814, 8.310653737682253320525659080767, 9.795807214726484063011577533994, 10.49008286057694862017779987876, 11.68985244330520288100802287367

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