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Properties

Label 2-888-37.7-c1-0-2
Degree $2$
Conductor $888$
Sign $0.546 - 0.837i$
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  + (−0.939 − 0.342i)3-s + (−0.482 − 2.73i)5-s + (0.268 + 1.52i)7-s + (0.766 + 0.642i)9-s + (−2.47 + 4.28i)11-s + (−2.48 + 2.08i)13-s + (−0.482 + 2.73i)15-s + (−1.92 − 1.61i)17-s + (6.30 + 2.29i)19-s + (0.268 − 1.52i)21-s + (0.772 + 1.33i)23-s + (−2.56 + 0.933i)25-s + (−0.500 − 0.866i)27-s + (−2.83 + 4.90i)29-s + 5.79·31-s + ⋯
L(s)  = 1  + (−0.542 − 0.197i)3-s + (−0.215 − 1.22i)5-s + (0.101 + 0.575i)7-s + (0.255 + 0.214i)9-s + (−0.746 + 1.29i)11-s + (−0.688 + 0.577i)13-s + (−0.124 + 0.706i)15-s + (−0.468 − 0.392i)17-s + (1.44 + 0.526i)19-s + (0.0585 − 0.332i)21-s + (0.161 + 0.279i)23-s + (−0.513 + 0.186i)25-s + (−0.0962 − 0.166i)27-s + (−0.525 + 0.910i)29-s + 1.04·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.546 - 0.837i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.546 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.812459 + 0.440105i\)
\(L(\frac12)\) \(\approx\) \(0.812459 + 0.440105i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (-3.05 - 5.25i)T \)
good5 \( 1 + (0.482 + 2.73i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.268 - 1.52i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (2.47 - 4.28i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.48 - 2.08i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.92 + 1.61i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-6.30 - 2.29i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-0.772 - 1.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.83 - 4.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.79T + 31T^{2} \)
41 \( 1 + (-4.71 + 3.95i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 1.60T + 43T^{2} \)
47 \( 1 + (0.689 + 1.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.934 - 5.30i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-0.121 + 0.689i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (9.43 - 7.91i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.54 - 14.4i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-3.59 - 1.30i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 1.58T + 73T^{2} \)
79 \( 1 + (-1.65 - 9.41i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.52 + 2.12i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (1.76 - 10.0i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (7.88 + 13.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.06234455879783456337105962443, −9.479654212065847080123955214425, −8.630219315101381557110015232483, −7.62191113434477973938520335806, −7.01394712130652909603980451045, −5.61599435132521135512301060777, −5.00823923400710692895628907395, −4.34537683259211983869541177607, −2.57317739869839364367664550856, −1.30172516819322606614808208812, 0.51700988750698519359478755328, 2.68135430029905847553271260135, 3.44635879751930593560043334834, 4.69265023005230295038727603618, 5.71879087918768457912323227011, 6.51572845971324465069007134378, 7.48209818396405597253789600859, 8.010778352678125175215429760435, 9.390890253051706872018580287147, 10.21517312114039444160894435462

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