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Properties

Label 2-888-37.34-c1-0-0
Degree $2$
Conductor $888$
Sign $-0.743 - 0.668i$
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  + (0.766 − 0.642i)3-s + (−2.56 − 0.933i)5-s + (−1.16 − 0.425i)7-s + (0.173 − 0.984i)9-s + (−1.02 + 1.77i)11-s + (−0.543 − 3.08i)13-s + (−2.56 + 0.933i)15-s + (−1.15 + 6.53i)17-s + (−2.48 + 2.08i)19-s + (−1.16 + 0.425i)21-s + (0.621 + 1.07i)23-s + (1.87 + 1.56i)25-s + (−0.500 − 0.866i)27-s + (−2.29 + 3.98i)29-s − 0.430·31-s + ⋯
L(s)  = 1  + (0.442 − 0.371i)3-s + (−1.14 − 0.417i)5-s + (−0.441 − 0.160i)7-s + (0.0578 − 0.328i)9-s + (−0.308 + 0.534i)11-s + (−0.150 − 0.854i)13-s + (−0.661 + 0.240i)15-s + (−0.279 + 1.58i)17-s + (−0.569 + 0.477i)19-s + (−0.254 + 0.0927i)21-s + (0.129 + 0.224i)23-s + (0.374 + 0.313i)25-s + (−0.0962 − 0.166i)27-s + (−0.426 + 0.739i)29-s − 0.0773·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.743 - 0.668i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ -0.743 - 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0531493 + 0.138576i\)
\(L(\frac12)\) \(\approx\) \(0.0531493 + 0.138576i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (5.89 - 1.49i)T \)
good5 \( 1 + (2.56 + 0.933i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.16 + 0.425i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.02 - 1.77i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.543 + 3.08i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.15 - 6.53i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (2.48 - 2.08i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-0.621 - 1.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.29 - 3.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.430T + 31T^{2} \)
41 \( 1 + (-0.763 - 4.32i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 7.14T + 43T^{2} \)
47 \( 1 + (-0.610 - 1.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.90 - 2.51i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-2.29 + 0.836i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.85 + 10.5i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-11.1 - 4.04i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-4.61 + 3.87i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + 4.77T + 73T^{2} \)
79 \( 1 + (-3.91 - 1.42i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.09 + 6.21i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (12.0 - 4.39i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-4.10 - 7.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.43553497640450353053482765964, −9.601601406743750927598417173896, −8.343437145321967962198829667225, −8.174470375508498739981655421915, −7.19249136956874518453652450971, −6.30092352221276360796857529794, −5.06519434861698181863034470959, −3.97187212290620806926998902266, −3.25414945589963242382528573651, −1.71548112029955252450560029268, 0.06437243109948729594848654782, 2.43278740174366127968062175569, 3.36509658927422193339209692131, 4.24507473997156569386159031003, 5.20992819367409688741271770688, 6.61021181795063050026310221901, 7.25963303663012867277250972762, 8.157693115832635438802445072053, 8.995396502044066639926928886493, 9.669736415752750355151941418068

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