Properties

Label 2-888-296.43-c1-0-7
Degree $2$
Conductor $888$
Sign $0.945 - 0.325i$
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  + (0.0571 − 1.41i)2-s − i·3-s + (−1.99 − 0.161i)4-s + (−0.322 + 0.322i)5-s + (−1.41 − 0.0571i)6-s + 0.857i·7-s + (−0.342 + 2.80i)8-s − 9-s + (0.437 + 0.474i)10-s + 1.26i·11-s + (−0.161 + 1.99i)12-s + (−2.24 + 2.24i)13-s + (1.21 + 0.0489i)14-s + (0.322 + 0.322i)15-s + (3.94 + 0.644i)16-s + (−1.69 − 1.69i)17-s + ⋯
L(s)  = 1  + (0.0404 − 0.999i)2-s − 0.577i·3-s + (−0.996 − 0.0807i)4-s + (−0.144 + 0.144i)5-s + (−0.576 − 0.0233i)6-s + 0.323i·7-s + (−0.120 + 0.992i)8-s − 0.333·9-s + (0.138 + 0.150i)10-s + 0.382i·11-s + (−0.0466 + 0.575i)12-s + (−0.623 + 0.623i)13-s + (0.323 + 0.0130i)14-s + (0.0833 + 0.0833i)15-s + (0.986 + 0.161i)16-s + (−0.411 − 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.945 - 0.325i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.945 - 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.826462 + 0.138150i\)
\(L(\frac12)\) \(\approx\) \(0.826462 + 0.138150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0571 + 1.41i)T \)
3 \( 1 + iT \)
37 \( 1 + (-1.50 + 5.89i)T \)
good5 \( 1 + (0.322 - 0.322i)T - 5iT^{2} \)
7 \( 1 - 0.857iT - 7T^{2} \)
11 \( 1 - 1.26iT - 11T^{2} \)
13 \( 1 + (2.24 - 2.24i)T - 13iT^{2} \)
17 \( 1 + (1.69 + 1.69i)T + 17iT^{2} \)
19 \( 1 + (-1.36 - 1.36i)T + 19iT^{2} \)
23 \( 1 + (5.38 - 5.38i)T - 23iT^{2} \)
29 \( 1 + (-4.56 - 4.56i)T + 29iT^{2} \)
31 \( 1 + (-3.15 - 3.15i)T + 31iT^{2} \)
41 \( 1 - 7.84iT - 41T^{2} \)
43 \( 1 + (3.36 + 3.36i)T + 43iT^{2} \)
47 \( 1 - 1.36iT - 47T^{2} \)
53 \( 1 + 9.41iT - 53T^{2} \)
59 \( 1 + (9.64 + 9.64i)T + 59iT^{2} \)
61 \( 1 + (-2.02 - 2.02i)T + 61iT^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 + 8.52iT - 71T^{2} \)
73 \( 1 - 8.93iT - 73T^{2} \)
79 \( 1 + (10.1 - 10.1i)T - 79iT^{2} \)
83 \( 1 + 3.47T + 83T^{2} \)
89 \( 1 + (-1.69 + 1.69i)T - 89iT^{2} \)
97 \( 1 + (-6.72 - 6.72i)T + 97iT^{2} \)
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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

−10.12343366792039705733439569489, −9.494502048479550452239021894648, −8.650938303109750143517607325805, −7.72156978781370304058690597727, −6.84770753168781485312071923190, −5.62427263696464504517088217512, −4.74872526076536206198356262216, −3.59352996128244788578532936466, −2.50540150994357387834863999186, −1.49211096168921837311537445445, 0.40986617500176209461009061862, 2.81680194509886233167196212078, 4.17442496012613465422780759838, 4.66187326035178288050824742172, 5.84528406472164585626063096493, 6.49743097879762078296219499016, 7.69292125794777744656743138194, 8.281349366158795483854611313705, 9.071941728935813449086340192138, 10.13229324823308563103054671718

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