Properties

Label 2-888-296.269-c1-0-65
Degree $2$
Conductor $888$
Sign $-0.965 + 0.260i$
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  + (−1.41 − 0.00150i)2-s + (0.866 − 0.5i)3-s + (1.99 + 0.00425i)4-s + (−1.17 + 0.680i)5-s + (−1.22 + 0.705i)6-s + (−0.930 − 1.61i)7-s + (−2.82 − 0.00902i)8-s + (0.499 − 0.866i)9-s + (1.66 − 0.961i)10-s − 5.10i·11-s + (1.73 − 0.996i)12-s + (−0.598 + 0.345i)13-s + (1.31 + 2.28i)14-s + (−0.680 + 1.17i)15-s + (3.99 + 0.0170i)16-s + (−2.81 + 4.88i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.00106i)2-s + (0.499 − 0.288i)3-s + (0.999 + 0.00212i)4-s + (−0.527 + 0.304i)5-s + (−0.500 + 0.288i)6-s + (−0.351 − 0.609i)7-s + (−0.999 − 0.00318i)8-s + (0.166 − 0.288i)9-s + (0.527 − 0.303i)10-s − 1.53i·11-s + (0.500 − 0.287i)12-s + (−0.166 + 0.0958i)13-s + (0.351 + 0.609i)14-s + (−0.175 + 0.304i)15-s + (0.999 + 0.00425i)16-s + (−0.683 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.965 + 0.260i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ -0.965 + 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0476836 - 0.359384i\)
\(L(\frac12)\) \(\approx\) \(0.0476836 - 0.359384i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.00150i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (5.59 + 2.38i)T \)
good5 \( 1 + (1.17 - 0.680i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.930 + 1.61i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 5.10iT - 11T^{2} \)
13 \( 1 + (0.598 - 0.345i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.81 - 4.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.01 + 1.74i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.73T + 23T^{2} \)
29 \( 1 - 0.624iT - 29T^{2} \)
31 \( 1 + 2.96T + 31T^{2} \)
41 \( 1 + (1.48 + 2.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.0870iT - 43T^{2} \)
47 \( 1 + 4.40T + 47T^{2} \)
53 \( 1 + (-0.220 - 0.127i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.3 + 6.55i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.79 - 2.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.62 + 5.55i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.16 - 8.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.235T + 73T^{2} \)
79 \( 1 + (5.43 + 9.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.23 + 4.17i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.86 + 4.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−9.660796917190545948411900043744, −8.756700840313355237183451637394, −8.146665001762721114212180381653, −7.38684261703321932054995521035, −6.58748320820477980615114409512, −5.73760372514469221976196030325, −3.81855728014519114067746649780, −3.20865118313842496669329044914, −1.78450035161272763093358882583, −0.21135758152152590867651748772, 1.87808076800280272458807491070, 2.84063856774445251467227464959, 4.14199137246041406632919037952, 5.25363876625260585298531013739, 6.51941712779401374928458694264, 7.42344297311505981226584154418, 8.027584017372207366435838607851, 8.954181969634185513455335524229, 9.673857414646652890068787004761, 10.06007145113300138926520052676

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