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Properties

Label 2-888-111.8-c1-0-35
Degree $2$
Conductor $888$
Sign $-0.998 + 0.0471i$
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  + (0.996 − 1.41i)3-s + (−0.601 − 2.24i)5-s + (−0.786 + 1.36i)7-s + (−1.01 − 2.82i)9-s − 0.738·11-s + (−5.22 + 1.40i)13-s + (−3.77 − 1.38i)15-s + (−6.88 − 1.84i)17-s + (−2.79 + 0.749i)19-s + (1.14 + 2.47i)21-s + (5.90 − 5.90i)23-s + (−0.345 + 0.199i)25-s + (−5.00 − 1.38i)27-s + (5.65 + 5.65i)29-s + (−1.19 + 1.19i)31-s + ⋯
L(s)  = 1  + (0.575 − 0.817i)3-s + (−0.268 − 1.00i)5-s + (−0.297 + 0.514i)7-s + (−0.337 − 0.941i)9-s − 0.222·11-s + (−1.45 + 0.388i)13-s + (−0.975 − 0.357i)15-s + (−1.66 − 0.447i)17-s + (−0.641 + 0.172i)19-s + (0.249 + 0.539i)21-s + (1.23 − 1.23i)23-s + (−0.0691 + 0.0399i)25-s + (−0.963 − 0.265i)27-s + (1.05 + 1.05i)29-s + (−0.215 + 0.215i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.998 + 0.0471i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ -0.998 + 0.0471i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0202434 - 0.857910i\)
\(L(\frac12)\) \(\approx\) \(0.0202434 - 0.857910i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.996 + 1.41i)T \)
37 \( 1 + (-4.55 + 4.03i)T \)
good5 \( 1 + (0.601 + 2.24i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.786 - 1.36i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.738T + 11T^{2} \)
13 \( 1 + (5.22 - 1.40i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (6.88 + 1.84i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.79 - 0.749i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.90 + 5.90i)T - 23iT^{2} \)
29 \( 1 + (-5.65 - 5.65i)T + 29iT^{2} \)
31 \( 1 + (1.19 - 1.19i)T - 31iT^{2} \)
41 \( 1 + (4.27 - 7.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.64 + 7.64i)T + 43iT^{2} \)
47 \( 1 + 10.6iT - 47T^{2} \)
53 \( 1 + (-4.00 + 2.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.81 + 0.755i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.69 + 10.0i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (5.21 + 3.00i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.51 - 1.44i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 + (1.59 - 0.426i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-12.2 + 7.06i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.62 + 13.5i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.824 - 0.824i)T + 97iT^{2} \)
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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

−9.321524898966652709445938937128, −8.793106024110671814436471227083, −8.273454501873667429775280237363, −7.00002632934522954589631840825, −6.62212310282219539432280459755, −5.12093566671538621465076324588, −4.46772063871493438058467496617, −2.89629219778701196858499038338, −2.04594268768854606602385964331, −0.35071677855560893783045949115, 2.41184787441965779745189760734, 3.11634038329858854875207579978, 4.23599014832022227720384452152, 4.99015641488797859117650654307, 6.40713927813496331406925879979, 7.22893887945317998756726394276, 7.953806363315390074463149753188, 9.018040375570101552917113001259, 9.776336887600676109564230563178, 10.56992055211189647989773418944

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