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Properties

Label 2-888-111.14-c1-0-34
Degree $2$
Conductor $888$
Sign $-0.758 - 0.651i$
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  + (−0.898 − 1.48i)3-s + (0.719 − 2.68i)5-s + (2.26 + 3.91i)7-s + (−1.38 + 2.66i)9-s − 4.17·11-s + (−6.14 − 1.64i)13-s + (−4.62 + 1.34i)15-s + (−5.34 + 1.43i)17-s + (1.15 + 0.310i)19-s + (3.76 − 6.86i)21-s + (0.554 + 0.554i)23-s + (−2.35 − 1.36i)25-s + (5.18 − 0.334i)27-s + (−4.45 + 4.45i)29-s + (−5.75 − 5.75i)31-s + ⋯
L(s)  = 1  + (−0.518 − 0.855i)3-s + (0.321 − 1.20i)5-s + (0.854 + 1.47i)7-s + (−0.462 + 0.886i)9-s − 1.25·11-s + (−1.70 − 0.456i)13-s + (−1.19 + 0.347i)15-s + (−1.29 + 0.347i)17-s + (0.265 + 0.0711i)19-s + (0.822 − 1.49i)21-s + (0.115 + 0.115i)23-s + (−0.471 − 0.272i)25-s + (0.997 − 0.0644i)27-s + (−0.828 + 0.828i)29-s + (−1.03 − 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.758 - 0.651i$
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: \((2,\ 888,\ (\ :1/2),\ -0.758 - 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0235861 + 0.0636509i\)
\(L(\frac12)\) \(\approx\) \(0.0235861 + 0.0636509i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.898 + 1.48i)T \)
37 \( 1 + (1.51 + 5.89i)T \)
good5 \( 1 + (-0.719 + 2.68i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.26 - 3.91i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.17T + 11T^{2} \)
13 \( 1 + (6.14 + 1.64i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (5.34 - 1.43i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.15 - 0.310i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.554 - 0.554i)T + 23iT^{2} \)
29 \( 1 + (4.45 - 4.45i)T - 29iT^{2} \)
31 \( 1 + (5.75 + 5.75i)T + 31iT^{2} \)
41 \( 1 + (4.00 + 6.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.15 + 2.15i)T - 43iT^{2} \)
47 \( 1 - 6.01iT - 47T^{2} \)
53 \( 1 + (4.08 + 2.35i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.458 + 0.122i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.89 - 7.05i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.496 + 0.286i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.25 + 0.725i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.0iT - 73T^{2} \)
79 \( 1 + (-6.35 - 1.70i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-5.56 - 3.21i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.508 + 1.89i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-11.4 + 11.4i)T - 97iT^{2} \)
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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

−9.351809310252162707234939470707, −8.734042086832323956381425471748, −7.915162315883238924505703473776, −7.23240528063165223863535888104, −5.69449290060782952782291950396, −5.34982986006023270037727564782, −4.75835578838984527566985229399, −2.45110437587285984645748319975, −1.91451502504182279543560092523, −0.03112532131319274798061776638, 2.26114296200379442940230114978, 3.40320391651284108043956041000, 4.68301413282098205215416650484, 5.01807251300012973149065209247, 6.50349533453984773461459102870, 7.16630056258360104767011976720, 7.87798301724780040891725847534, 9.298122559769736830398128181225, 10.11850233792591340713497188631, 10.59788723531268115900372072787

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