Properties

Label 2-888-111.14-c1-0-1
Degree $2$
Conductor $888$
Sign $-0.891 - 0.452i$
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  + (−1.64 − 0.544i)3-s + (−0.557 + 2.07i)5-s + (1.90 + 3.30i)7-s + (2.40 + 1.79i)9-s − 2.70·11-s + (0.737 + 0.197i)13-s + (2.04 − 3.11i)15-s + (−0.554 + 0.148i)17-s + (−2.06 − 0.552i)19-s + (−1.33 − 6.46i)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  + (−0.949 − 0.314i)3-s + (−0.249 + 0.930i)5-s + (0.720 + 1.24i)7-s + (0.802 + 0.597i)9-s − 0.815·11-s + (0.204 + 0.0547i)13-s + (0.529 − 0.804i)15-s + (−0.134 + 0.0360i)17-s + (−0.472 − 0.126i)19-s + (−0.291 − 1.41i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.891 - 0.452i$
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: \((2,\ 888,\ (\ :1/2),\ -0.891 - 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.138944 + 0.581023i\)
\(L(\frac12)\) \(\approx\) \(0.138944 + 0.581023i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 0.544i)T \)
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

−10.72269715880945505895904321591, −9.908356723393620066712209152166, −8.592746528875152153310944804722, −7.939011406135429639620117377679, −6.92874271342328561479642243510, −6.16596659694651722072187319841, −5.34385934749268046276385793492, −4.47910691377479210890708192466, −2.89208793765423947532756977612, −1.86797254552254682176266201502, 0.32358730309272099318224462974, 1.59909232259946103358567483753, 3.73739274778528385772302258514, 4.51771995459062677367513082448, 5.17554342355730979846351879476, 6.15458069500450558075287730482, 7.34175203370911298241868916025, 7.939921224842521158842065766808, 8.958546265379618928252192038361, 10.03707944206683593928204823733

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