Properties

Label 2-888-296.43-c1-0-30
Degree $2$
Conductor $888$
Sign $0.999 + 0.0254i$
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  + (−1.41 − 0.0802i)2-s − i·3-s + (1.98 + 0.226i)4-s + (0.922 − 0.922i)5-s + (−0.0802 + 1.41i)6-s + 1.23i·7-s + (−2.78 − 0.479i)8-s − 9-s + (−1.37 + 1.22i)10-s + 1.12i·11-s + (0.226 − 1.98i)12-s + (−1.45 + 1.45i)13-s + (0.0993 − 1.74i)14-s + (−0.922 − 0.922i)15-s + (3.89 + 0.900i)16-s + (2.73 + 2.73i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0567i)2-s − 0.577i·3-s + (0.993 + 0.113i)4-s + (0.412 − 0.412i)5-s + (−0.0327 + 0.576i)6-s + 0.468i·7-s + (−0.985 − 0.169i)8-s − 0.333·9-s + (−0.435 + 0.388i)10-s + 0.338i·11-s + (0.0653 − 0.573i)12-s + (−0.403 + 0.403i)13-s + (0.0265 − 0.467i)14-s + (−0.238 − 0.238i)15-s + (0.974 + 0.225i)16-s + (0.663 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.999 + 0.0254i$
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: \((2,\ 888,\ (\ :1/2),\ 0.999 + 0.0254i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07923 - 0.0137515i\)
\(L(\frac12)\) \(\approx\) \(1.07923 - 0.0137515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0802i)T \)
3 \( 1 + iT \)
37 \( 1 + (5.15 + 3.22i)T \)
good5 \( 1 + (-0.922 + 0.922i)T - 5iT^{2} \)
7 \( 1 - 1.23iT - 7T^{2} \)
11 \( 1 - 1.12iT - 11T^{2} \)
13 \( 1 + (1.45 - 1.45i)T - 13iT^{2} \)
17 \( 1 + (-2.73 - 2.73i)T + 17iT^{2} \)
19 \( 1 + (-4.43 - 4.43i)T + 19iT^{2} \)
23 \( 1 + (-5.25 + 5.25i)T - 23iT^{2} \)
29 \( 1 + (2.19 + 2.19i)T + 29iT^{2} \)
31 \( 1 + (-4.60 - 4.60i)T + 31iT^{2} \)
41 \( 1 - 2.84iT - 41T^{2} \)
43 \( 1 + (-1.25 - 1.25i)T + 43iT^{2} \)
47 \( 1 - 2.71iT - 47T^{2} \)
53 \( 1 + 8.17iT - 53T^{2} \)
59 \( 1 + (-0.580 - 0.580i)T + 59iT^{2} \)
61 \( 1 + (-2.21 - 2.21i)T + 61iT^{2} \)
67 \( 1 + 9.61iT - 67T^{2} \)
71 \( 1 + 8.74iT - 71T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + (-4.35 + 4.35i)T - 79iT^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + (5.93 - 5.93i)T - 89iT^{2} \)
97 \( 1 + (5.86 + 5.86i)T + 97iT^{2} \)
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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

−9.952675932642007516312984035777, −9.248691766538205736291036618721, −8.501332070510879236769434677541, −7.69949990587438400205499971806, −6.88389061072374038709071521858, −5.97706017171823174686648926084, −5.11139440705454474613602475708, −3.36061207465670527924667576805, −2.16426619346049025697424627149, −1.17627910584909565239200981921, 0.861691684875170251840468048137, 2.60991172666440068703443026861, 3.40157127779760400523593485253, 5.03206919823621882256025272295, 5.83279017747654544118284644301, 7.02897036896174178950331051024, 7.51397555957147333921401327354, 8.639772827000564646284204481152, 9.469197496399455455299500623624, 9.990835836280412214603866150797

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