Properties

Label 2-888-111.14-c1-0-18
Degree $2$
Conductor $888$
Sign $0.680 - 0.732i$
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  + (1.31 + 1.12i)3-s + (0.299 − 1.11i)5-s + (0.728 + 1.26i)7-s + (0.464 + 2.96i)9-s + 2.75·11-s + (5.87 + 1.57i)13-s + (1.65 − 1.13i)15-s + (−4.20 + 1.12i)17-s + (−5.86 − 1.57i)19-s + (−0.461 + 2.47i)21-s + (−4.77 − 4.77i)23-s + (3.16 + 1.82i)25-s + (−2.72 + 4.42i)27-s + (3.63 − 3.63i)29-s + (4.56 + 4.56i)31-s + ⋯
L(s)  = 1  + (0.759 + 0.650i)3-s + (0.134 − 0.500i)5-s + (0.275 + 0.476i)7-s + (0.154 + 0.987i)9-s + 0.830·11-s + (1.62 + 0.436i)13-s + (0.427 − 0.293i)15-s + (−1.02 + 0.273i)17-s + (−1.34 − 0.360i)19-s + (−0.100 + 0.541i)21-s + (−0.995 − 0.995i)23-s + (0.633 + 0.365i)25-s + (−0.524 + 0.851i)27-s + (0.674 − 0.674i)29-s + (0.819 + 0.819i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.680 - 0.732i$
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: \((2,\ 888,\ (\ :1/2),\ 0.680 - 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07622 + 0.905073i\)
\(L(\frac12)\) \(\approx\) \(2.07622 + 0.905073i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 - 1.12i)T \)
37 \( 1 + (-6.04 - 0.640i)T \)
good5 \( 1 + (-0.299 + 1.11i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.728 - 1.26i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.75T + 11T^{2} \)
13 \( 1 + (-5.87 - 1.57i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.20 - 1.12i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.86 + 1.57i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.77 + 4.77i)T + 23iT^{2} \)
29 \( 1 + (-3.63 + 3.63i)T - 29iT^{2} \)
31 \( 1 + (-4.56 - 4.56i)T + 31iT^{2} \)
41 \( 1 + (-1.37 - 2.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.55 + 2.55i)T - 43iT^{2} \)
47 \( 1 - 9.51iT - 47T^{2} \)
53 \( 1 + (7.21 + 4.16i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.55 + 0.953i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.66 - 13.6i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-4.17 + 2.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.53 - 4.92i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.05iT - 73T^{2} \)
79 \( 1 + (12.9 + 3.46i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (3.62 + 2.09i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.69 + 10.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.19 + 1.19i)T - 97iT^{2} \)
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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

−10.20850750846461997024117195620, −9.050326905229648447061712599423, −8.710457974589663990911865165471, −8.197085509718573773801813591107, −6.64387939696849263756856247250, −5.98091697381753662692920647588, −4.42681793600101569790767462569, −4.24889729339720408399948154241, −2.74786906435215636263738457449, −1.59332319626564896009168292464, 1.17621887672581887230399184930, 2.36118572388792203918891059981, 3.60388721538992653863866655872, 4.30911396428669435458494105585, 6.19019110344285594410326380786, 6.44259885339122693529891416670, 7.53647706789072672482404057875, 8.383272259359275543687775799758, 8.938839595165369349773204365370, 9.998857886301522298291261854162

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