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Properties

Label 2-888-111.14-c1-0-10
Degree $2$
Conductor $888$
Sign $0.117 - 0.993i$
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  + (1.70 − 0.287i)3-s + (−0.721 + 2.69i)5-s + (2.47 + 4.29i)7-s + (2.83 − 0.980i)9-s − 1.92·11-s + (1.17 + 0.315i)13-s + (−0.459 + 4.80i)15-s + (−5.26 + 1.40i)17-s + (−4.56 − 1.22i)19-s + (5.46 + 6.62i)21-s + (−2.41 − 2.41i)23-s + (−2.39 − 1.38i)25-s + (4.56 − 2.48i)27-s + (3.35 − 3.35i)29-s + (2.96 + 2.96i)31-s + ⋯
L(s)  = 1  + (0.986 − 0.165i)3-s + (−0.322 + 1.20i)5-s + (0.936 + 1.62i)7-s + (0.945 − 0.326i)9-s − 0.579·11-s + (0.326 + 0.0875i)13-s + (−0.118 + 1.24i)15-s + (−1.27 + 0.341i)17-s + (−1.04 − 0.280i)19-s + (1.19 + 1.44i)21-s + (−0.503 − 0.503i)23-s + (−0.479 − 0.276i)25-s + (0.877 − 0.478i)27-s + (0.622 − 0.622i)29-s + (0.532 + 0.532i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.117 - 0.993i$
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: \((2,\ 888,\ (\ :1/2),\ 0.117 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58177 + 1.40529i\)
\(L(\frac12)\) \(\approx\) \(1.58177 + 1.40529i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.287i)T \)
37 \( 1 + (5.71 + 2.08i)T \)
good5 \( 1 + (0.721 - 2.69i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.47 - 4.29i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.92T + 11T^{2} \)
13 \( 1 + (-1.17 - 0.315i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (5.26 - 1.40i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.56 + 1.22i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.41 + 2.41i)T + 23iT^{2} \)
29 \( 1 + (-3.35 + 3.35i)T - 29iT^{2} \)
31 \( 1 + (-2.96 - 2.96i)T + 31iT^{2} \)
41 \( 1 + (-2.27 - 3.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.86 + 3.86i)T - 43iT^{2} \)
47 \( 1 + 1.47iT - 47T^{2} \)
53 \( 1 + (-11.0 - 6.37i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.7 + 2.87i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.65 + 9.89i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-8.89 + 5.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.72 + 0.996i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.224iT - 73T^{2} \)
79 \( 1 + (-7.22 - 1.93i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-13.1 - 7.56i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.85 + 6.91i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.0 - 10.0i)T - 97iT^{2} \)
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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

−10.47002290891735225007251519418, −9.240841398100838976656380254880, −8.444987188018685303505997288187, −8.118532021099228186777674536570, −6.90112632390782572353363522512, −6.23906559036219148582899428274, −4.92008559688720210765981684946, −3.83769721469835888030390418623, −2.45044170937677825559182756924, −2.27080134396899201786019221138, 0.928409286159755764897349092733, 2.16972787793105250736851252041, 3.87490059932220516772705663180, 4.35597102063116247775824476229, 5.12507460716381336791494190433, 6.82310020175561961567951611243, 7.62019326005337105267570170370, 8.415233202350099742301567489230, 8.726788224605121453172161168149, 9.992608142894837919012817517939

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