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Properties

Label 2-888-111.8-c1-0-22
Degree $2$
Conductor $888$
Sign $0.323 + 0.946i$
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  + (−1.42 − 0.981i)3-s + (−0.212 − 0.791i)5-s + (−0.149 + 0.258i)7-s + (1.07 + 2.80i)9-s + 1.40·11-s + (2.07 − 0.556i)13-s + (−0.474 + 1.33i)15-s + (4.03 + 1.08i)17-s + (−4.30 + 1.15i)19-s + (0.466 − 0.222i)21-s + (3.40 − 3.40i)23-s + (3.74 − 2.16i)25-s + (1.21 − 5.05i)27-s + (−2.74 − 2.74i)29-s + (−0.659 + 0.659i)31-s + ⋯
L(s)  = 1  + (−0.823 − 0.566i)3-s + (−0.0948 − 0.353i)5-s + (−0.0563 + 0.0976i)7-s + (0.357 + 0.933i)9-s + 0.423·11-s + (0.575 − 0.154i)13-s + (−0.122 + 0.345i)15-s + (0.978 + 0.262i)17-s + (−0.987 + 0.264i)19-s + (0.101 − 0.0485i)21-s + (0.709 − 0.709i)23-s + (0.749 − 0.432i)25-s + (0.234 − 0.972i)27-s + (−0.509 − 0.509i)29-s + (−0.118 + 0.118i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.323 + 0.946i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.323 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.932729 - 0.666930i\)
\(L(\frac12)\) \(\approx\) \(0.932729 - 0.666930i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 + 0.981i)T \)
37 \( 1 + (3.30 - 5.10i)T \)
good5 \( 1 + (0.212 + 0.791i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.149 - 0.258i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.40T + 11T^{2} \)
13 \( 1 + (-2.07 + 0.556i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.03 - 1.08i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.30 - 1.15i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.40 + 3.40i)T - 23iT^{2} \)
29 \( 1 + (2.74 + 2.74i)T + 29iT^{2} \)
31 \( 1 + (0.659 - 0.659i)T - 31iT^{2} \)
41 \( 1 + (-5.90 + 10.2i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.89 + 4.89i)T + 43iT^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 + (-6.63 + 3.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.53 + 0.410i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.777 + 2.89i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.51 + 0.876i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.84 + 1.64i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.542iT - 73T^{2} \)
79 \( 1 + (-4.66 + 1.25i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-7.59 + 4.38i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.75 + 6.56i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.62 - 7.62i)T + 97iT^{2} \)
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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

−10.30050801680302800421703661924, −8.937468468024474010412238320411, −8.309741617155465545144100356853, −7.28351555558119940349528936460, −6.45892552459626531831865804440, −5.68490437175496791938983731357, −4.77314583155875044191777600059, −3.65820144249366712043588433709, −2.05367257210922205383313421129, −0.74391181968017738048426866658, 1.19902680700939438088855879724, 3.11219382164208766520684891188, 4.01731948542766680640832237822, 5.02834141755174273636096608881, 5.95359585183855662576910317697, 6.73791751776090655503901177989, 7.58573197579442707616729426178, 8.886734694343509331237716006672, 9.486376274964251585910394134514, 10.46505279485074921480648483081

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