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Properties

Label 2-888-296.51-c1-0-64
Degree $2$
Conductor $888$
Sign $0.146 + 0.989i$
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 + i)2-s + (0.866 − 0.5i)3-s − 2i·4-s + (0.5 − 1.86i)5-s + (−0.366 + 1.36i)6-s + (0.633 − 0.366i)7-s + (2 + 2i)8-s + (0.499 − 0.866i)9-s + (1.36 + 2.36i)10-s − 4.73i·11-s + (−1 − 1.73i)12-s + (0.0980 − 0.366i)13-s + (−0.267 + i)14-s + (−0.5 − 1.86i)15-s − 4·16-s + (−0.964 − 3.59i)17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.499 − 0.288i)3-s − i·4-s + (0.223 − 0.834i)5-s + (−0.149 + 0.557i)6-s + (0.239 − 0.138i)7-s + (0.707 + 0.707i)8-s + (0.166 − 0.288i)9-s + (0.431 + 0.748i)10-s − 1.42i·11-s + (−0.288 − 0.499i)12-s + (0.0272 − 0.101i)13-s + (−0.0716 + 0.267i)14-s + (−0.129 − 0.481i)15-s − 16-s + (−0.233 − 0.872i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.146 + 0.989i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.146 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881800 - 0.760986i\)
\(L(\frac12)\) \(\approx\) \(0.881800 - 0.760986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (5.5 + 2.59i)T \)
good5 \( 1 + (-0.5 + 1.86i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.633 + 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.73iT - 11T^{2} \)
13 \( 1 + (-0.0980 + 0.366i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.964 + 3.59i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.09 + 1.09i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.73 - 2.73i)T + 23iT^{2} \)
29 \( 1 + (3.83 - 3.83i)T - 29iT^{2} \)
31 \( 1 + (7.19 - 7.19i)T - 31iT^{2} \)
41 \( 1 + (-8.13 + 4.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 + i)T - 43iT^{2} \)
47 \( 1 + 8.19iT - 47T^{2} \)
53 \( 1 + (0.803 + 0.464i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.66 + 13.6i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.86 - 1.03i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-9.63 + 5.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.90 + 2.83i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (1.63 - 6.09i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (5.46 - 9.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9.69 + 2.59i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.75 + 9.75i)T - 97iT^{2} \)
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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

−9.475773398476375755316758960012, −8.880796323797135999238593183755, −8.496747983335346288367262241105, −7.46029460108141030230264953896, −6.72661380272397349640672628163, −5.57778135041578771462886876280, −4.96731613885297824787956272225, −3.46717010769558827824476328048, −1.89114638629302314907524361158, −0.65726831908913632077006705882, 1.88985027937771945718824631724, 2.53920315434645911183358823310, 3.83286917255509148762851299788, 4.58616282073920657708067197744, 6.24084805301408981054731056991, 7.19724022784264638706496152743, 7.893307641211519072194397503011, 8.845908321243155492042828473719, 9.568398892276509931279195587661, 10.32896566173124592427127470026

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