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Properties

Label 2-888-37.33-c1-0-9
Degree $2$
Conductor $888$
Sign $0.893 - 0.449i$
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  + (0.173 + 0.984i)3-s + (0.835 − 0.700i)5-s + (2.72 − 2.28i)7-s + (−0.939 + 0.342i)9-s + (−3.18 + 5.52i)11-s + (4.45 + 1.62i)13-s + (0.835 + 0.700i)15-s + (1.18 − 0.430i)17-s + (−1.07 − 6.11i)19-s + (2.72 + 2.28i)21-s + (1.77 + 3.07i)23-s + (−0.661 + 3.75i)25-s + (−0.5 − 0.866i)27-s + (2.53 − 4.39i)29-s + 7.64·31-s + ⋯
L(s)  = 1  + (0.100 + 0.568i)3-s + (0.373 − 0.313i)5-s + (1.02 − 0.862i)7-s + (−0.313 + 0.114i)9-s + (−0.961 + 1.66i)11-s + (1.23 + 0.449i)13-s + (0.215 + 0.180i)15-s + (0.286 − 0.104i)17-s + (−0.247 − 1.40i)19-s + (0.593 + 0.498i)21-s + (0.370 + 0.641i)23-s + (−0.132 + 0.750i)25-s + (−0.0962 − 0.166i)27-s + (0.471 − 0.816i)29-s + 1.37·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.893 - 0.449i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.893 - 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88921 + 0.448520i\)
\(L(\frac12)\) \(\approx\) \(1.88921 + 0.448520i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-1.18 + 5.96i)T \)
good5 \( 1 + (-0.835 + 0.700i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-2.72 + 2.28i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (3.18 - 5.52i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.45 - 1.62i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.18 + 0.430i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (1.07 + 6.11i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-1.77 - 3.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.53 + 4.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.64T + 31T^{2} \)
41 \( 1 + (-11.5 - 4.19i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 7.19T + 43T^{2} \)
47 \( 1 + (-1.43 - 2.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.53 + 3.80i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.325 - 0.273i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.76 - 1.00i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (11.2 - 9.42i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.08 - 11.8i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 - 8.57T + 73T^{2} \)
79 \( 1 + (7.70 - 6.46i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.76 - 1.36i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (6.47 + 5.43i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.03 - 1.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.08449092161095415218233951046, −9.520718649658708092064121146150, −8.524281210973944651394727530714, −7.69184721400272194143786213746, −6.92441776585961637766865603100, −5.60098952605787631674247748336, −4.65463373508567208036435744384, −4.23156199570498903790170604991, −2.62147393145799535449876172236, −1.33615024336394533897683859746, 1.15882601067304442138272185639, 2.48638610791957252952707692099, 3.38336095884338403603869589702, 4.97320744589848771910196599787, 5.98780747684731416871196330476, 6.22879688686721906346411507703, 7.964356197725983902931342306361, 8.220893260468821799334894773465, 8.867050459973243137354685835604, 10.38578131653027980800100151530

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