Properties

Label 2-888-111.14-c1-0-11
Degree $2$
Conductor $888$
Sign $0.345 - 0.938i$
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  + (−0.350 + 1.69i)3-s + (0.557 − 2.07i)5-s + (1.90 + 3.30i)7-s + (−2.75 − 1.18i)9-s + 2.70·11-s + (0.737 + 0.197i)13-s + (3.33 + 1.67i)15-s + (0.554 − 0.148i)17-s + (−2.06 − 0.552i)19-s + (−6.26 + 2.07i)21-s + (4.13 + 4.13i)23-s + (0.315 + 0.181i)25-s + (2.98 − 4.25i)27-s + (1.48 − 1.48i)29-s + (0.369 + 0.369i)31-s + ⋯
L(s)  = 1  + (−0.202 + 0.979i)3-s + (0.249 − 0.930i)5-s + (0.720 + 1.24i)7-s + (−0.918 − 0.396i)9-s + 0.815·11-s + (0.204 + 0.0547i)13-s + (0.860 + 0.432i)15-s + (0.134 − 0.0360i)17-s + (−0.472 − 0.126i)19-s + (−1.36 + 0.453i)21-s + (0.862 + 0.862i)23-s + (0.0630 + 0.0363i)25-s + (0.573 − 0.819i)27-s + (0.275 − 0.275i)29-s + (0.0663 + 0.0663i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.345 - 0.938i$
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: \((2,\ 888,\ (\ :1/2),\ 0.345 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36255 + 0.949938i\)
\(L(\frac12)\) \(\approx\) \(1.36255 + 0.949938i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.350 - 1.69i)T \)
37 \( 1 + (1.29 - 5.94i)T \)
good5 \( 1 + (-0.557 + 2.07i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.90 - 3.30i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.70T + 11T^{2} \)
13 \( 1 + (-0.737 - 0.197i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.554 + 0.148i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.06 + 0.552i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.13 - 4.13i)T + 23iT^{2} \)
29 \( 1 + (-1.48 + 1.48i)T - 29iT^{2} \)
31 \( 1 + (-0.369 - 0.369i)T + 31iT^{2} \)
41 \( 1 + (-1.11 - 1.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.18 - 5.18i)T - 43iT^{2} \)
47 \( 1 - 1.34iT - 47T^{2} \)
53 \( 1 + (-0.415 - 0.239i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.4 + 3.34i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.788 + 2.94i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.47 - 1.42i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.44 - 3.14i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.3iT - 73T^{2} \)
79 \( 1 + (10.2 + 2.75i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-4.79 - 2.76i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.12 + 11.6i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.215 + 0.215i)T - 97iT^{2} \)
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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

−10.11506282904238538047133157473, −9.316347772994000706715738881438, −8.769168893750113610796723633423, −8.221906776312775371892997226681, −6.62781527333372118551740917185, −5.60744966476866406616482773226, −5.06115239401482809898238240523, −4.22003967034764089793823640837, −2.91964976773045251686069151397, −1.43110800541654201087662309987, 0.943672902737671596452416739801, 2.16028450788360531612824403551, 3.45042206935676898932361423771, 4.59884992286900428104159659192, 5.84733340112985743245847600923, 6.88814136676295859158231991375, 7.05541356393905242603469186198, 8.134656305641662630783652896784, 8.943038369080521869935058360531, 10.37760730367178986959719256840

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