Properties

Label 2-888-111.14-c1-0-24
Degree $2$
Conductor $888$
Sign $-0.114 + 0.993i$
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  + (−1.50 − 0.855i)3-s + (1.03 − 3.84i)5-s + (−0.379 − 0.657i)7-s + (1.53 + 2.57i)9-s + 3.35·11-s + (6.41 + 1.71i)13-s + (−4.84 + 4.91i)15-s + (5.90 − 1.58i)17-s + (2.47 + 0.663i)19-s + (0.00890 + 1.31i)21-s + (−2.09 − 2.09i)23-s + (−9.40 − 5.43i)25-s + (−0.105 − 5.19i)27-s + (0.264 − 0.264i)29-s + (−2.04 − 2.04i)31-s + ⋯
L(s)  = 1  + (−0.869 − 0.494i)3-s + (0.461 − 1.72i)5-s + (−0.143 − 0.248i)7-s + (0.511 + 0.859i)9-s + 1.01·11-s + (1.77 + 0.476i)13-s + (−1.25 + 1.26i)15-s + (1.43 − 0.384i)17-s + (0.568 + 0.152i)19-s + (0.00194 + 0.286i)21-s + (−0.436 − 0.436i)23-s + (−1.88 − 1.08i)25-s + (−0.0203 − 0.999i)27-s + (0.0491 − 0.0491i)29-s + (−0.367 − 0.367i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.114 + 0.993i$
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: \((2,\ 888,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.984041 - 1.10413i\)
\(L(\frac12)\) \(\approx\) \(0.984041 - 1.10413i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 + 0.855i)T \)
37 \( 1 + (5.73 - 2.01i)T \)
good5 \( 1 + (-1.03 + 3.84i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.379 + 0.657i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.35T + 11T^{2} \)
13 \( 1 + (-6.41 - 1.71i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-5.90 + 1.58i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.47 - 0.663i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.09 + 2.09i)T + 23iT^{2} \)
29 \( 1 + (-0.264 + 0.264i)T - 29iT^{2} \)
31 \( 1 + (2.04 + 2.04i)T + 31iT^{2} \)
41 \( 1 + (2.58 + 4.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.470 + 0.470i)T - 43iT^{2} \)
47 \( 1 - 13.4iT - 47T^{2} \)
53 \( 1 + (-4.60 - 2.66i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (13.7 - 3.69i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.70 - 6.35i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (4.28 - 2.47i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.31 + 4.22i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.99iT - 73T^{2} \)
79 \( 1 + (2.73 + 0.733i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-2.14 - 1.23i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.69 + 13.7i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.60 - 7.60i)T - 97iT^{2} \)
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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

−9.828156538753507016849153516447, −9.038474075589196505866623862270, −8.319972719075522787333699049938, −7.31359065338849045018482716836, −6.11708520931996501218179548813, −5.71783281201853835853131578260, −4.67328234082072337968323627345, −3.77094657869260524393820353361, −1.53695887251261399968054010073, −0.997449352758469216028906423977, 1.48214175874389396883157971816, 3.40177968766569945134864898082, 3.61752178722334699990693522754, 5.42053859797430410112152018283, 6.09902444556899795487028289294, 6.61024358069765574806154394159, 7.58942312467910037765812660296, 8.888620121393357784854843367339, 9.843201134441345018070449353053, 10.39186064973639058898531568058

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