Properties

Label 2-888-888.443-c1-0-91
Degree $2$
Conductor $888$
Sign $0.918 + 0.394i$
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  + (−1.39 + 0.249i)2-s + (1.03 + 1.38i)3-s + (1.87 − 0.693i)4-s + 0.556i·5-s + (−1.79 − 1.66i)6-s − 3.66i·7-s + (−2.43 + 1.43i)8-s + (−0.837 + 2.88i)9-s + (−0.138 − 0.774i)10-s − 4.20i·11-s + (2.91 + 1.87i)12-s − 2.20·13-s + (0.914 + 5.10i)14-s + (−0.770 + 0.578i)15-s + (3.03 − 2.60i)16-s + 3.13·17-s + ⋯
L(s)  = 1  + (−0.984 + 0.176i)2-s + (0.600 + 0.799i)3-s + (0.937 − 0.346i)4-s + 0.248i·5-s + (−0.731 − 0.681i)6-s − 1.38i·7-s + (−0.862 + 0.506i)8-s + (−0.279 + 0.960i)9-s + (−0.0438 − 0.244i)10-s − 1.26i·11-s + (0.840 + 0.541i)12-s − 0.612·13-s + (0.244 + 1.36i)14-s + (−0.199 + 0.149i)15-s + (0.759 − 0.650i)16-s + 0.761·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.918 + 0.394i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.918 + 0.394i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10803 - 0.227680i\)
\(L(\frac12)\) \(\approx\) \(1.10803 - 0.227680i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.249i)T \)
3 \( 1 + (-1.03 - 1.38i)T \)
37 \( 1 + (-6.08 - 0.0599i)T \)
good5 \( 1 - 0.556iT - 5T^{2} \)
7 \( 1 + 3.66iT - 7T^{2} \)
11 \( 1 + 4.20iT - 11T^{2} \)
13 \( 1 + 2.20T + 13T^{2} \)
17 \( 1 - 3.13T + 17T^{2} \)
19 \( 1 - 1.33iT - 19T^{2} \)
23 \( 1 + 6.03iT - 23T^{2} \)
29 \( 1 + 8.61iT - 29T^{2} \)
31 \( 1 - 2.20T + 31T^{2} \)
41 \( 1 - 5.29iT - 41T^{2} \)
43 \( 1 + 7.58iT - 43T^{2} \)
47 \( 1 - 5.37T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 + 5.05T + 59T^{2} \)
61 \( 1 - 1.07T + 61T^{2} \)
67 \( 1 - 0.856T + 67T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 - 0.501T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 7.18iT - 83T^{2} \)
89 \( 1 + 3.48T + 89T^{2} \)
97 \( 1 - 7.96iT - 97T^{2} \)
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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

−9.988860106590999977687537671629, −9.398877034969087568289962660614, −8.217089246790656350623412453061, −7.929903559079482510880915854034, −6.87144500799220927084856241540, −5.90591669127794472254348470155, −4.62356623905799345584013031990, −3.49955841423481814950482830754, −2.56162234275392741690407275374, −0.73436058885315683468102051288, 1.41564192452905874956778474206, 2.38087132385115288859456944366, 3.22852846800109236202363600515, 5.03850788454410117755148206149, 6.13481646401904570188986571224, 7.11384288077762552367683050916, 7.71997627715535782925925546800, 8.580059223507376783781389163973, 9.376316274190768424664217873136, 9.671665747245545076171491121867

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