Properties

Label 2-888-111.14-c1-0-15
Degree $2$
Conductor $888$
Sign $0.864 - 0.502i$
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  + (1.63 − 0.557i)3-s + (−0.799 + 2.98i)5-s + (−0.335 − 0.580i)7-s + (2.37 − 1.82i)9-s + 4.05·11-s + (−0.293 − 0.0785i)13-s + (0.353 + 5.33i)15-s + (−0.650 + 0.174i)17-s + (0.739 + 0.198i)19-s + (−0.873 − 0.765i)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  + (0.946 − 0.322i)3-s + (−0.357 + 1.33i)5-s + (−0.126 − 0.219i)7-s + (0.792 − 0.609i)9-s + 1.22·11-s + (−0.0812 − 0.0217i)13-s + (0.0911 + 1.37i)15-s + (−0.157 + 0.0422i)17-s + (0.169 + 0.0454i)19-s + (−0.190 − 0.167i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.864 - 0.502i$
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: \((2,\ 888,\ (\ :1/2),\ 0.864 - 0.502i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10488 + 0.567792i\)
\(L(\frac12)\) \(\approx\) \(2.10488 + 0.567792i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 0.557i)T \)
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

−10.06814557967335689563037948362, −9.397355658022641118373677046132, −8.448178824543840835677372696452, −7.58450523818499266577696738318, −6.79158205483769952646647273421, −6.37807145466402098877722408250, −4.59032929107661856932252482928, −3.47747597871951956467243090920, −2.96736880523698494273612999098, −1.50896203994373773319911332403, 1.11797412034877175963753514779, 2.50345849367804918515480361690, 3.92590265806942313986929744980, 4.40954309972902912971454712032, 5.50164173010758376486958403766, 6.74989870228410990879292637774, 7.79946180148203490452667451446, 8.531371773378725560605850099221, 9.209678960813067842042424127857, 9.609921050535093516014199740884

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