Properties

Label 2-888-111.14-c1-0-28
Degree $2$
Conductor $888$
Sign $-0.709 + 0.704i$
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  + (−0.546 + 1.64i)3-s + (0.245 − 0.916i)5-s + (−0.268 − 0.464i)7-s + (−2.40 − 1.79i)9-s + 2.15·11-s + (−4.73 − 1.26i)13-s + (1.37 + 0.904i)15-s + (−7.20 + 1.93i)17-s + (−0.618 − 0.165i)19-s + (0.909 − 0.186i)21-s + (−3.45 − 3.45i)23-s + (3.55 + 2.05i)25-s + (4.26 − 2.96i)27-s + (−4.81 + 4.81i)29-s + (−5.47 − 5.47i)31-s + ⋯
L(s)  = 1  + (−0.315 + 0.948i)3-s + (0.109 − 0.409i)5-s + (−0.101 − 0.175i)7-s + (−0.800 − 0.599i)9-s + 0.649·11-s + (−1.31 − 0.351i)13-s + (0.354 + 0.233i)15-s + (−1.74 + 0.468i)17-s + (−0.141 − 0.0380i)19-s + (0.198 − 0.0407i)21-s + (−0.720 − 0.720i)23-s + (0.710 + 0.410i)25-s + (0.821 − 0.570i)27-s + (−0.894 + 0.894i)29-s + (−0.984 − 0.984i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.709 + 0.704i$
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: \((2,\ 888,\ (\ :1/2),\ -0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0673210 - 0.163221i\)
\(L(\frac12)\) \(\approx\) \(0.0673210 - 0.163221i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.546 - 1.64i)T \)
37 \( 1 + (3.08 - 5.24i)T \)
good5 \( 1 + (-0.245 + 0.916i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.268 + 0.464i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
13 \( 1 + (4.73 + 1.26i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (7.20 - 1.93i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.618 + 0.165i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.45 + 3.45i)T + 23iT^{2} \)
29 \( 1 + (4.81 - 4.81i)T - 29iT^{2} \)
31 \( 1 + (5.47 + 5.47i)T + 31iT^{2} \)
41 \( 1 + (2.52 + 4.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.21 + 5.21i)T - 43iT^{2} \)
47 \( 1 - 9.16iT - 47T^{2} \)
53 \( 1 + (6.26 + 3.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.47 - 2.54i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.71 + 13.8i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-8.60 + 4.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.24 - 2.44i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.462iT - 73T^{2} \)
79 \( 1 + (-6.71 - 1.80i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-1.63 - 0.944i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.448 + 1.67i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.43 - 9.43i)T - 97iT^{2} \)
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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

−9.698897364953424483455977780633, −9.163148523032609227778052757223, −8.398295587099045601493641832733, −7.12490406656949979149348608270, −6.27541796241271302115901431611, −5.20570375142283372838997780365, −4.49803126460676095286649406134, −3.59620568169349415898847948718, −2.16530269714524405597013672520, −0.080599496364857643653555982144, 1.85638754847391591000935110867, 2.70440134132486041724518904689, 4.24201619296018787698265802226, 5.32947920484463635933614686628, 6.35197368935124095327491095112, 6.98625527975977797273706824187, 7.64191068459675893882903842374, 8.836862606183956432355922903332, 9.455567441022960229782909247400, 10.62611778392989620293041758361

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