Properties

Label 2-888-111.14-c1-0-13
Degree $2$
Conductor $888$
Sign $0.945 - 0.325i$
Analytic cond. $7.09071$
Root an. cond. $2.66283$
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  + (−0.709 + 1.58i)3-s + (−0.440 + 1.64i)5-s + (−2.41 − 4.18i)7-s + (−1.99 − 2.24i)9-s + 5.79·11-s + (0.132 + 0.0354i)13-s + (−2.28 − 1.86i)15-s + (1.95 − 0.523i)17-s + (2.26 + 0.605i)19-s + (8.31 − 0.849i)21-s + (−2.57 − 2.57i)23-s + (1.82 + 1.05i)25-s + (4.95 − 1.56i)27-s + (1.61 − 1.61i)29-s + (2.85 + 2.85i)31-s + ⋯
L(s)  = 1  + (−0.409 + 0.912i)3-s + (−0.196 + 0.734i)5-s + (−0.912 − 1.58i)7-s + (−0.664 − 0.747i)9-s + 1.74·11-s + (0.0367 + 0.00984i)13-s + (−0.589 − 0.480i)15-s + (0.473 − 0.126i)17-s + (0.518 + 0.139i)19-s + (1.81 − 0.185i)21-s + (−0.536 − 0.536i)23-s + (0.364 + 0.210i)25-s + (0.953 − 0.300i)27-s + (0.299 − 0.299i)29-s + (0.512 + 0.512i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.945 - 0.325i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.945 - 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23337 + 0.206638i\)
\(L(\frac12)\) \(\approx\) \(1.23337 + 0.206638i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.709 - 1.58i)T \)
37 \( 1 + (4.69 - 3.86i)T \)
good5 \( 1 + (0.440 - 1.64i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.41 + 4.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.79T + 11T^{2} \)
13 \( 1 + (-0.132 - 0.0354i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.95 + 0.523i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.26 - 0.605i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.57 + 2.57i)T + 23iT^{2} \)
29 \( 1 + (-1.61 + 1.61i)T - 29iT^{2} \)
31 \( 1 + (-2.85 - 2.85i)T + 31iT^{2} \)
41 \( 1 + (-2.81 - 4.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.06 + 7.06i)T - 43iT^{2} \)
47 \( 1 + 7.92iT - 47T^{2} \)
53 \( 1 + (-4.24 - 2.45i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.36 + 1.16i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.81 - 14.2i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.58 - 0.913i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.1 + 5.85i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.5iT - 73T^{2} \)
79 \( 1 + (6.87 + 1.84i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-3.79 - 2.19i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.55 - 5.81i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.1 + 12.1i)T - 97iT^{2} \)
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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

−10.25425071173996301744842129677, −9.604750396942754249866044775572, −8.728094752050135825270828670399, −7.31985207180059531464575035990, −6.71664302327830066324962111180, −6.02582150727785545840837044960, −4.54732065460934885380243786587, −3.77705434459040236949762927291, −3.21562434747760085782749167711, −0.863481484091512628367670416606, 1.03797238207066478756987303120, 2.33942812490743196734245820129, 3.60423894774403074106208183479, 5.01068794103825742432290556928, 5.96398177097805239494112808860, 6.40094657955215783908674555988, 7.48975891199197306464615459655, 8.582312556548943574324953349452, 9.066854987360290215933318305328, 9.833383264161676437832389497395

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