Properties

Label 2-888-888.443-c1-0-19
Degree $2$
Conductor $888$
Sign $0.249 - 0.968i$
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 + (−1.55 + 1.88i)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 + (−0.698 + 3.39i)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 + (−0.636 + 0.771i)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 + (−0.201 + 0.979i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.249 - 0.968i$
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: \((2,\ 888,\ (\ :1/2),\ 0.249 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15157 + 0.892010i\)
\(L(\frac12)\) \(\approx\) \(1.15157 + 0.892010i\)
\(L(\frac{3}{2})\) 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

−10.30672896856960579132608268342, −9.778746270508648494563973699838, −9.001558249097744487468693887870, −7.43103673891095334045606559922, −6.56110977704680929520964655868, −5.65451435588228534387262817333, −4.96602591083851232824067193549, −4.35729767016126533041626771852, −2.81671023819199920359330656674, −1.80636047776295137045689937890, 0.56297263003048972787591646600, 2.52159890504658353092237626568, 3.86732820211811399454986135609, 4.72174085644805048789405072415, 5.53877891758909905769441326771, 6.54696984131030892280030341004, 7.27807075886569494242879186664, 7.62744698969824958275595930300, 9.019508635911622077269920344853, 10.37772242916402624250596510314

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