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Properties

Label 2-888-111.14-c1-0-31
Degree $2$
Conductor $888$
Sign $-0.695 + 0.718i$
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  + (0.925 − 1.46i)3-s + (0.476 − 1.77i)5-s + (−2.08 − 3.61i)7-s + (−1.28 − 2.70i)9-s + 2.49·11-s + (5.42 + 1.45i)13-s + (−2.16 − 2.34i)15-s + (−4.54 + 1.21i)17-s + (3.13 + 0.840i)19-s + (−7.21 − 0.288i)21-s + (−1.79 − 1.79i)23-s + (1.39 + 0.807i)25-s + (−5.15 − 0.621i)27-s + (−6.12 + 6.12i)29-s + (−3.95 − 3.95i)31-s + ⋯
L(s)  = 1  + (0.534 − 0.845i)3-s + (0.212 − 0.794i)5-s + (−0.787 − 1.36i)7-s + (−0.429 − 0.903i)9-s + 0.753·11-s + (1.50 + 0.402i)13-s + (−0.558 − 0.604i)15-s + (−1.10 + 0.295i)17-s + (0.719 + 0.192i)19-s + (−1.57 − 0.0629i)21-s + (−0.375 − 0.375i)23-s + (0.279 + 0.161i)25-s + (−0.992 − 0.119i)27-s + (−1.13 + 1.13i)29-s + (−0.710 − 0.710i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.695 + 0.718i$
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: \((2,\ 888,\ (\ :1/2),\ -0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.680132 - 1.60416i\)
\(L(\frac12)\) \(\approx\) \(0.680132 - 1.60416i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.925 + 1.46i)T \)
37 \( 1 + (-1.62 + 5.86i)T \)
good5 \( 1 + (-0.476 + 1.77i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.08 + 3.61i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.49T + 11T^{2} \)
13 \( 1 + (-5.42 - 1.45i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.54 - 1.21i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.13 - 0.840i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.79 + 1.79i)T + 23iT^{2} \)
29 \( 1 + (6.12 - 6.12i)T - 29iT^{2} \)
31 \( 1 + (3.95 + 3.95i)T + 31iT^{2} \)
41 \( 1 + (2.65 + 4.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.38 - 5.38i)T - 43iT^{2} \)
47 \( 1 + 5.49iT - 47T^{2} \)
53 \( 1 + (-10.7 - 6.21i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.91 + 1.04i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.537 - 2.00i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-12.0 + 6.93i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-13.3 + 7.70i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 + (9.49 + 2.54i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-6.83 - 3.94i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.26 - 8.44i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.136 + 0.136i)T - 97iT^{2} \)
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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

−9.474701334741232732641730654500, −9.013981945355409258868354054937, −8.209469014922715922180891261757, −7.06863169006887489519191974591, −6.65053363302020340423516542435, −5.63090374631842654962190375582, −4.00059758837196415016496652927, −3.60227563201341941319793272718, −1.80493792166872356725789817453, −0.807813469896483850740962124954, 2.16615349204374954293050496853, 3.14606217646247530310020190030, 3.83549827521207783713691302672, 5.27356879461286405918895863713, 6.09345253563088022470594807978, 6.84209137868450157521549552463, 8.246207150942734985974760799867, 8.915861721048139451069722677635, 9.514439735931465880997552265842, 10.27621957076525719667155857390

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