Properties

Label 2-888-111.14-c1-0-36
Degree $2$
Conductor $888$
Sign $-0.607 + 0.794i$
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  + (1.30 − 1.14i)3-s + (0.799 − 2.98i)5-s + (−0.335 − 0.580i)7-s + (0.395 − 2.97i)9-s − 4.05·11-s + (−0.293 − 0.0785i)13-s + (−2.36 − 4.79i)15-s + (0.650 − 0.174i)17-s + (0.739 + 0.198i)19-s + (−1.09 − 0.374i)21-s + (−2.25 − 2.25i)23-s + (−3.92 − 2.26i)25-s + (−2.87 − 4.32i)27-s + (1.54 − 1.54i)29-s + (6.22 + 6.22i)31-s + ⋯
L(s)  = 1  + (0.752 − 0.658i)3-s + (0.357 − 1.33i)5-s + (−0.126 − 0.219i)7-s + (0.131 − 0.991i)9-s − 1.22·11-s + (−0.0812 − 0.0217i)13-s + (−0.610 − 1.23i)15-s + (0.157 − 0.0422i)17-s + (0.169 + 0.0454i)19-s + (−0.240 − 0.0816i)21-s + (−0.469 − 0.469i)23-s + (−0.785 − 0.453i)25-s + (−0.554 − 0.832i)27-s + (0.287 − 0.287i)29-s + (1.11 + 1.11i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.607 + 0.794i$
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: \((2,\ 888,\ (\ :1/2),\ -0.607 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804042 - 1.62607i\)
\(L(\frac12)\) \(\approx\) \(0.804042 - 1.62607i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 1.14i)T \)
37 \( 1 + (-0.816 - 6.02i)T \)
good5 \( 1 + (-0.799 + 2.98i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.335 + 0.580i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.05T + 11T^{2} \)
13 \( 1 + (0.293 + 0.0785i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.650 + 0.174i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.739 - 0.198i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.25 + 2.25i)T + 23iT^{2} \)
29 \( 1 + (-1.54 + 1.54i)T - 29iT^{2} \)
31 \( 1 + (-6.22 - 6.22i)T + 31iT^{2} \)
41 \( 1 + (3.27 + 5.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.22 - 3.22i)T - 43iT^{2} \)
47 \( 1 - 0.410iT - 47T^{2} \)
53 \( 1 + (-5.53 - 3.19i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.82 + 2.63i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.33 + 4.96i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-3.94 + 2.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.45 - 5.45i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.9iT - 73T^{2} \)
79 \( 1 + (6.36 + 1.70i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-8.30 - 4.79i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.326 - 1.21i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-10.9 + 10.9i)T - 97iT^{2} \)
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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

−9.809905155179367130913586293783, −8.682382153315951997342543131786, −8.356997190421199814398061028562, −7.46372910716683475130638059533, −6.44182760044178602121020870084, −5.36659264987100591304993571195, −4.53311047370442820771968224257, −3.19166886561575948328288157076, −2.05030360833605159711131296405, −0.77807363121018911161962723882, 2.31200538006318917505740583731, 2.87363208702999846688610184164, 3.91593324339948735126868773234, 5.15839149674170338692234622854, 6.05541502555109773954839484397, 7.19681093048422248800861827208, 7.87183981754632279852967988732, 8.777199545028321372222458403106, 9.944433203423506971019671198486, 10.14430695877329506757028837769

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