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Properties

Label 2-888-37.7-c1-0-16
Degree $2$
Conductor $888$
Sign $-0.631 + 0.775i$
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 + (0.139 + 0.793i)5-s + (−0.580 − 3.29i)7-s + (0.766 + 0.642i)9-s + (−2.31 + 4.01i)11-s + (3.55 − 2.98i)13-s + (0.139 − 0.793i)15-s + (−3.21 − 2.70i)17-s + (−1.11 − 0.406i)19-s + (−0.580 + 3.29i)21-s + (−1.67 − 2.89i)23-s + (4.08 − 1.48i)25-s + (−0.500 − 0.866i)27-s + (−0.410 + 0.710i)29-s − 10.7·31-s + ⋯
L(s)  = 1  + (−0.542 − 0.197i)3-s + (0.0625 + 0.354i)5-s + (−0.219 − 1.24i)7-s + (0.255 + 0.214i)9-s + (−0.698 + 1.20i)11-s + (0.985 − 0.827i)13-s + (0.0361 − 0.204i)15-s + (−0.780 − 0.654i)17-s + (−0.255 − 0.0931i)19-s + (−0.126 + 0.717i)21-s + (−0.348 − 0.603i)23-s + (0.817 − 0.297i)25-s + (−0.0962 − 0.166i)27-s + (−0.0761 + 0.131i)29-s − 1.92·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.631 + 0.775i$
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: \((2,\ 888,\ (\ :1/2),\ -0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303145 - 0.637380i\)
\(L(\frac12)\) \(\approx\) \(0.303145 - 0.637380i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.49 + 5.54i)T \)
good5 \( 1 + (-0.139 - 0.793i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.580 + 3.29i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (2.31 - 4.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.55 + 2.98i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (3.21 + 2.70i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (1.11 + 0.406i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (1.67 + 2.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.410 - 0.710i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
41 \( 1 + (3.23 - 2.71i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 4.12T + 43T^{2} \)
47 \( 1 + (6.37 + 11.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.807 + 4.57i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-0.449 + 2.54i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (4.78 - 4.01i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (2.54 + 14.4i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-11.0 - 4.01i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 9.68T + 73T^{2} \)
79 \( 1 + (2.45 + 13.9i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.98 - 2.50i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (0.904 - 5.13i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-4.48 - 7.77i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.18343299781996494651935968088, −9.055521230662672904138020701520, −7.904477515761004951173702362344, −7.11833955255895518444046464150, −6.61137864669296206515754774219, −5.39953036601898315156200917439, −4.49429389964487140969459542948, −3.45412262871672899211688228300, −2.01558026175829111353004000240, −0.35424335450922232747783177456, 1.65223024048429996448772823056, 3.07707396941853316418751787199, 4.20105568413993608032319986698, 5.45288040710486383884774812437, 5.88100370418439239321481971453, 6.76115736801076191153671127207, 8.165917821351403326884702883636, 8.851732090276741381893529921784, 9.358197084820567942326042538802, 10.70684321827884742049872676291

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