Properties

Label 2-888-111.14-c1-0-12
Degree $2$
Conductor $888$
Sign $0.784 + 0.620i$
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  + (−1.69 − 0.348i)3-s + (−0.245 + 0.916i)5-s + (−0.268 − 0.464i)7-s + (2.75 + 1.18i)9-s − 2.15·11-s + (−4.73 − 1.26i)13-s + (0.735 − 1.46i)15-s + (7.20 − 1.93i)17-s + (−0.618 − 0.165i)19-s + (0.293 + 0.881i)21-s + (3.45 + 3.45i)23-s + (3.55 + 2.05i)25-s + (−4.26 − 2.96i)27-s + (4.81 − 4.81i)29-s + (−5.47 − 5.47i)31-s + ⋯
L(s)  = 1  + (−0.979 − 0.200i)3-s + (−0.109 + 0.409i)5-s + (−0.101 − 0.175i)7-s + (0.919 + 0.393i)9-s − 0.649·11-s + (−1.31 − 0.351i)13-s + (0.189 − 0.379i)15-s + (1.74 − 0.468i)17-s + (−0.141 − 0.0380i)19-s + (0.0640 + 0.192i)21-s + (0.720 + 0.720i)23-s + (0.710 + 0.410i)25-s + (−0.821 − 0.570i)27-s + (0.894 − 0.894i)29-s + (−0.984 − 0.984i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.784 + 0.620i$
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: \((2,\ 888,\ (\ :1/2),\ 0.784 + 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898915 - 0.312693i\)
\(L(\frac12)\) \(\approx\) \(0.898915 - 0.312693i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.348i)T \)
37 \( 1 + (3.08 - 5.24i)T \)
good5 \( 1 + (0.245 - 0.916i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.268 + 0.464i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.15T + 11T^{2} \)
13 \( 1 + (4.73 + 1.26i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-7.20 + 1.93i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.618 + 0.165i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.45 - 3.45i)T + 23iT^{2} \)
29 \( 1 + (-4.81 + 4.81i)T - 29iT^{2} \)
31 \( 1 + (5.47 + 5.47i)T + 31iT^{2} \)
41 \( 1 + (-2.52 - 4.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.21 + 5.21i)T - 43iT^{2} \)
47 \( 1 + 9.16iT - 47T^{2} \)
53 \( 1 + (-6.26 - 3.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.47 + 2.54i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.71 + 13.8i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-8.60 + 4.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.24 + 2.44i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.462iT - 73T^{2} \)
79 \( 1 + (-6.71 - 1.80i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (1.63 + 0.944i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.448 - 1.67i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.43 - 9.43i)T - 97iT^{2} \)
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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

−10.08833105285105711799679811329, −9.593564476768132139953299972799, −8.013275871343570447070754379642, −7.38758310194249932461765065060, −6.72403309303129173803628984913, −5.40742099635523794117048094557, −5.12468233782965740826195632080, −3.65426412581245756684194336231, −2.42828883717369320009758016166, −0.67520504414020987748709920669, 1.01513093039895788573043266162, 2.73147494714696334863129300557, 4.13098462942261985714046083527, 5.12229754064439775483870602613, 5.57836039259473369016030659487, 6.83774132991334364580119036981, 7.48560678051110789073905562784, 8.617068803020784534338248889717, 9.524435054726407449503849769232, 10.38699071813549976728865754921

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