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Properties

Label 2-888-111.14-c1-0-35
Degree $2$
Conductor $888$
Sign $-0.989 + 0.142i$
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  + (1.03 − 1.38i)3-s + (−0.272 + 1.01i)5-s + (−0.338 − 0.586i)7-s + (−0.853 − 2.87i)9-s − 6.32·11-s + (−3.74 − 1.00i)13-s + (1.13 + 1.43i)15-s + (−2.40 + 0.644i)17-s + (−5.39 − 1.44i)19-s + (−1.16 − 0.137i)21-s + (−0.0101 − 0.0101i)23-s + (3.36 + 1.94i)25-s + (−4.87 − 1.79i)27-s + (4.64 − 4.64i)29-s + (−1.74 − 1.74i)31-s + ⋯
L(s)  = 1  + (0.598 − 0.801i)3-s + (−0.121 + 0.455i)5-s + (−0.127 − 0.221i)7-s + (−0.284 − 0.958i)9-s − 1.90·11-s + (−1.03 − 0.278i)13-s + (0.291 + 0.370i)15-s + (−0.583 + 0.156i)17-s + (−1.23 − 0.331i)19-s + (−0.254 − 0.0299i)21-s + (−0.00210 − 0.00210i)23-s + (0.673 + 0.388i)25-s + (−0.938 − 0.345i)27-s + (0.863 − 0.863i)29-s + (−0.312 − 0.312i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.989 + 0.142i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ -0.989 + 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0435110 - 0.607704i\)
\(L(\frac12)\) \(\approx\) \(0.0435110 - 0.607704i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 + 1.38i)T \)
37 \( 1 + (-4.57 + 4.00i)T \)
good5 \( 1 + (0.272 - 1.01i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.338 + 0.586i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 6.32T + 11T^{2} \)
13 \( 1 + (3.74 + 1.00i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.40 - 0.644i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.39 + 1.44i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.0101 + 0.0101i)T + 23iT^{2} \)
29 \( 1 + (-4.64 + 4.64i)T - 29iT^{2} \)
31 \( 1 + (1.74 + 1.74i)T + 31iT^{2} \)
41 \( 1 + (-2.38 - 4.13i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.64 + 4.64i)T - 43iT^{2} \)
47 \( 1 + 2.77iT - 47T^{2} \)
53 \( 1 + (9.01 + 5.20i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.306 + 0.0820i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.49 - 5.56i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (7.70 - 4.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.32 + 2.49i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 16.2iT - 73T^{2} \)
79 \( 1 + (2.45 + 0.658i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-3.12 - 1.80i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.49 - 5.56i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.85 - 4.85i)T - 97iT^{2} \)
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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

−9.779040025897313928296802268170, −8.676972725312042242601016948479, −7.922392929205607732659976149130, −7.29854278461119847788334124623, −6.49998528574377679943164078646, −5.39849309272155834985809873550, −4.22847737531700940356618800362, −2.78113386624928997369416870263, −2.34824815492030209079550724449, −0.24122553158371505678805690504, 2.29581988992404780081479664127, 3.01074306437890529448925239133, 4.62476896651439520246840086274, 4.81242924779506335213695467157, 6.06178954547562736991326487258, 7.41944889553846033841479765624, 8.137106385916230183921217441335, 8.866531043822706496144899826622, 9.643317751370761754003175063725, 10.58381902346492772805433237957

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