Properties

Label 2-888-111.14-c1-0-22
Degree $2$
Conductor $888$
Sign $0.986 + 0.164i$
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  + (1.73 − 0.0692i)3-s + (−0.476 + 1.77i)5-s + (−2.08 − 3.61i)7-s + (2.99 − 0.239i)9-s − 2.49·11-s + (5.42 + 1.45i)13-s + (−0.700 + 3.10i)15-s + (4.54 − 1.21i)17-s + (3.13 + 0.840i)19-s + (−3.85 − 6.10i)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  + (0.999 − 0.0399i)3-s + (−0.212 + 0.794i)5-s + (−0.787 − 1.36i)7-s + (0.996 − 0.0798i)9-s − 0.753·11-s + (1.50 + 0.402i)13-s + (−0.180 + 0.802i)15-s + (1.10 − 0.295i)17-s + (0.719 + 0.192i)19-s + (−0.841 − 1.33i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.986 + 0.164i$
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: \((2,\ 888,\ (\ :1/2),\ 0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14045 - 0.177213i\)
\(L(\frac12)\) \(\approx\) \(2.14045 - 0.177213i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.0692i)T \)
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

−9.977602120111866333451429156841, −9.492713120142873157589576367231, −8.210895483970628606958797352773, −7.57749772831926198222221452838, −6.92748028434767174334393282177, −5.99322391299261142542901119053, −4.38198282095432597721641132931, −3.46442386660816235158383983310, −2.96889541167161710266335073229, −1.17404942145148247728056733633, 1.33721004288245283288172733151, 2.90321617931795790301764651510, 3.41664981166320804594083125661, 4.89633756309690257261917562015, 5.67611317096562800654225193001, 6.76014038981917446834105928005, 7.972760957211641926859875773431, 8.633619774756265558913914597122, 8.991537152006421058825857018451, 9.989459044554863985158333839907

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