Properties

Label 2-888-111.14-c1-0-17
Degree $2$
Conductor $888$
Sign $0.728 - 0.685i$
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  + (1.36 + 1.06i)3-s + (0.356 − 1.32i)5-s + (0.659 + 1.14i)7-s + (0.736 + 2.90i)9-s − 0.353·11-s + (−0.106 − 0.0285i)13-s + (1.90 − 1.43i)15-s + (−0.268 + 0.0720i)17-s + (7.97 + 2.13i)19-s + (−0.313 + 2.26i)21-s + (−0.593 − 0.593i)23-s + (2.68 + 1.55i)25-s + (−2.08 + 4.75i)27-s + (−1.88 + 1.88i)29-s + (3.87 + 3.87i)31-s + ⋯
L(s)  = 1  + (0.789 + 0.614i)3-s + (0.159 − 0.594i)5-s + (0.249 + 0.431i)7-s + (0.245 + 0.969i)9-s − 0.106·11-s + (−0.0295 − 0.00792i)13-s + (0.491 − 0.371i)15-s + (−0.0651 + 0.0174i)17-s + (1.82 + 0.489i)19-s + (−0.0684 + 0.494i)21-s + (−0.123 − 0.123i)23-s + (0.537 + 0.310i)25-s + (−0.401 + 0.915i)27-s + (−0.350 + 0.350i)29-s + (0.696 + 0.696i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.728 - 0.685i$
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: \((2,\ 888,\ (\ :1/2),\ 0.728 - 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05924 + 0.816272i\)
\(L(\frac12)\) \(\approx\) \(2.05924 + 0.816272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 1.06i)T \)
37 \( 1 + (5.93 + 1.32i)T \)
good5 \( 1 + (-0.356 + 1.32i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.659 - 1.14i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.353T + 11T^{2} \)
13 \( 1 + (0.106 + 0.0285i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.268 - 0.0720i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-7.97 - 2.13i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.593 + 0.593i)T + 23iT^{2} \)
29 \( 1 + (1.88 - 1.88i)T - 29iT^{2} \)
31 \( 1 + (-3.87 - 3.87i)T + 31iT^{2} \)
41 \( 1 + (-0.379 - 0.657i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.16 + 4.16i)T - 43iT^{2} \)
47 \( 1 + 6.18iT - 47T^{2} \)
53 \( 1 + (3.52 + 2.03i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.80 - 1.55i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.41 - 5.29i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-4.53 + 2.61i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.08 + 2.93i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.03iT - 73T^{2} \)
79 \( 1 + (-0.472 - 0.126i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (11.5 + 6.66i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.27 - 15.9i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.97 - 7.97i)T - 97iT^{2} \)
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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

−10.05952097610876482797666327486, −9.291918795471076409876378706964, −8.690403542801280180092838717420, −7.909055475093600755583338011617, −7.00430755160349375259860108006, −5.46334315806248348037660584418, −5.04129037725249338427043028255, −3.81656021660891955052544657972, −2.84826861786107456178527317917, −1.53689259320586156292986852169, 1.14063603455344475154905888365, 2.54678322344870836318118467989, 3.34961088048933733333301956244, 4.55122176512704625663302608895, 5.84667327105359132736600833513, 6.83286366369375133471788761300, 7.49125415318363302281029469695, 8.159490628836242399119887705670, 9.270679253277274379872583580206, 9.835667924803142714247217038844

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