Properties

Label 2-888-296.269-c1-0-45
Degree $2$
Conductor $888$
Sign $0.642 + 0.766i$
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  + (−1.40 + 0.139i)2-s + (−0.866 + 0.5i)3-s + (1.96 − 0.393i)4-s + (3.03 − 1.75i)5-s + (1.14 − 0.824i)6-s + (−0.822 − 1.42i)7-s + (−2.70 + 0.827i)8-s + (0.499 − 0.866i)9-s + (−4.03 + 2.89i)10-s + 2.11i·11-s + (−1.50 + 1.32i)12-s + (1.47 − 0.851i)13-s + (1.35 + 1.88i)14-s + (−1.75 + 3.03i)15-s + (3.69 − 1.54i)16-s + (2.13 − 3.69i)17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0987i)2-s + (−0.499 + 0.288i)3-s + (0.980 − 0.196i)4-s + (1.35 − 0.784i)5-s + (0.469 − 0.336i)6-s + (−0.310 − 0.538i)7-s + (−0.956 + 0.292i)8-s + (0.166 − 0.288i)9-s + (−1.27 + 0.914i)10-s + 0.637i·11-s + (−0.433 + 0.381i)12-s + (0.408 − 0.236i)13-s + (0.362 + 0.504i)14-s + (−0.452 + 0.784i)15-s + (0.922 − 0.385i)16-s + (0.517 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.956865 - 0.446571i\)
\(L(\frac12)\) \(\approx\) \(0.956865 - 0.446571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.139i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (-1.62 + 5.86i)T \)
good5 \( 1 + (-3.03 + 1.75i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.822 + 1.42i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.11iT - 11T^{2} \)
13 \( 1 + (-1.47 + 0.851i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.13 + 3.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.79 + 2.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.62T + 23T^{2} \)
29 \( 1 - 10.2iT - 29T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
41 \( 1 + (3.17 + 5.49i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 2.08iT - 43T^{2} \)
47 \( 1 - 0.00681T + 47T^{2} \)
53 \( 1 + (6.76 + 3.90i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.97 - 2.29i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.6 + 7.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.61 + 3.81i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.41 - 9.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.95T + 73T^{2} \)
79 \( 1 + (7.31 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.48 - 0.856i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.52 - 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.02T + 97T^{2} \)
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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

−9.779165719493857232657631643521, −9.397878737484141329228746768145, −8.672493978951196017005007457049, −7.27685059733980286616228139727, −6.80793213435095905040173723482, −5.44416316008353734572308840732, −5.26528994470627563622584253434, −3.43767669638588396552503343417, −1.94176554536960688579222577803, −0.815584951944564951285666704338, 1.36952849807433663305317952230, 2.40753605520553280964759835064, 3.46170929377004604454212682965, 5.64110280009563506589376141259, 5.99257298200646541493360845622, 6.71025755532699831751378635527, 7.74585731373088735204705838257, 8.665194557370379715874335821645, 9.731601843551945447371340785887, 9.966535685356263759797819554983

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