Properties

Label 2-888-888.443-c1-0-1
Degree $2$
Conductor $888$
Sign $-0.862 + 0.506i$
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  + (−1.24 + 0.661i)2-s + (−1.59 − 0.663i)3-s + (1.12 − 1.65i)4-s + 0.287i·5-s + (2.43 − 0.229i)6-s + 4.18i·7-s + (−0.308 + 2.81i)8-s + (2.11 + 2.12i)9-s + (−0.190 − 0.358i)10-s − 4.65i·11-s + (−2.89 + 1.90i)12-s − 5.63·13-s + (−2.76 − 5.22i)14-s + (0.190 − 0.459i)15-s + (−1.47 − 3.71i)16-s + 1.68·17-s + ⋯
L(s)  = 1  + (−0.883 + 0.468i)2-s + (−0.923 − 0.383i)3-s + (0.561 − 0.827i)4-s + 0.128i·5-s + (0.995 − 0.0935i)6-s + 1.58i·7-s + (−0.109 + 0.994i)8-s + (0.706 + 0.708i)9-s + (−0.0601 − 0.113i)10-s − 1.40i·11-s + (−0.836 + 0.548i)12-s − 1.56·13-s + (−0.740 − 1.39i)14-s + (0.0492 − 0.118i)15-s + (−0.368 − 0.929i)16-s + 0.408·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.862 + 0.506i$
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: \((2,\ 888,\ (\ :1/2),\ -0.862 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0102866 - 0.0378104i\)
\(L(\frac12)\) \(\approx\) \(0.0102866 - 0.0378104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 - 0.661i)T \)
3 \( 1 + (1.59 + 0.663i)T \)
37 \( 1 + (-5.22 - 3.11i)T \)
good5 \( 1 - 0.287iT - 5T^{2} \)
7 \( 1 - 4.18iT - 7T^{2} \)
11 \( 1 + 4.65iT - 11T^{2} \)
13 \( 1 + 5.63T + 13T^{2} \)
17 \( 1 - 1.68T + 17T^{2} \)
19 \( 1 - 5.89iT - 19T^{2} \)
23 \( 1 - 2.85iT - 23T^{2} \)
29 \( 1 + 4.39iT - 29T^{2} \)
31 \( 1 + 4.81T + 31T^{2} \)
41 \( 1 + 10.7iT - 41T^{2} \)
43 \( 1 + 2.32iT - 43T^{2} \)
47 \( 1 + 9.32T + 47T^{2} \)
53 \( 1 - 1.16T + 53T^{2} \)
59 \( 1 + 14.9T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + 9.33T + 67T^{2} \)
71 \( 1 + 3.72T + 71T^{2} \)
73 \( 1 + 0.925T + 73T^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 + 0.772iT - 83T^{2} \)
89 \( 1 + 8.30T + 89T^{2} \)
97 \( 1 + 15.3iT - 97T^{2} \)
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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

−10.54429243330094697649165888122, −9.734390383303479813644546010980, −8.912576975478597708377645851386, −8.014522007050359108524445250187, −7.32703622436732681112729956662, −6.10851209448543113475215550342, −5.77520334509645028123883118233, −4.98163276196419076765042487740, −2.88725765358215265511148548581, −1.71398294639901403986709636904, 0.02943089867976328556800850443, 1.36359891648719277736604550501, 2.98822231830311948446476804193, 4.48316530804070696907220363406, 4.73829009573187038839912700278, 6.60571685218941971275612091095, 7.18286007559605289428455510238, 7.68656215980896288964410960782, 9.303457319868946824162745089240, 9.714947221628626348666736207007

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