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Properties

Label 2-888-24.11-c1-0-50
Degree $2$
Conductor $888$
Sign $0.707 + 0.707i$
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  + (−0.366 − 1.36i)2-s − 1.73i·3-s + (−1.73 + i)4-s + 2.73·5-s + (−2.36 + 0.633i)6-s + 2i·7-s + (2 + 1.99i)8-s − 2.99·9-s + (−1 − 3.73i)10-s + 5.46i·11-s + (1.73 + 2.99i)12-s + 2i·13-s + (2.73 − 0.732i)14-s − 4.73i·15-s + (1.99 − 3.46i)16-s − 0.732i·17-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s − 0.999i·3-s + (−0.866 + 0.5i)4-s + 1.22·5-s + (−0.965 + 0.258i)6-s + 0.755i·7-s + (0.707 + 0.707i)8-s − 0.999·9-s + (−0.316 − 1.18i)10-s + 1.64i·11-s + (0.499 + 0.866i)12-s + 0.554i·13-s + (0.730 − 0.195i)14-s − 1.22i·15-s + (0.499 − 0.866i)16-s − 0.177i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37736 - 0.570523i\)
\(L(\frac12)\) \(\approx\) \(1.37736 - 0.570523i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
3 \( 1 + 1.73iT \)
37 \( 1 + iT \)
good5 \( 1 - 2.73T + 5T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 5.46iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 0.732iT - 17T^{2} \)
19 \( 1 - 4.19T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 0.196T + 29T^{2} \)
31 \( 1 - 6.19iT - 31T^{2} \)
41 \( 1 - 2.53iT - 41T^{2} \)
43 \( 1 + 6.73T + 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 3.46iT - 59T^{2} \)
61 \( 1 + 15.4iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 8.53T + 71T^{2} \)
73 \( 1 + 8.92T + 73T^{2} \)
79 \( 1 + 13.1iT - 79T^{2} \)
83 \( 1 - 6.39iT - 83T^{2} \)
89 \( 1 + 7.66iT - 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−9.852418554448503316260255209842, −9.331955795196998394329892898576, −8.645371590334883190033333761728, −7.45936220657401884451242389028, −6.73248930853009747340355172178, −5.50550686043735181835488245219, −4.82093989680765355779996452224, −3.05086586580877887217236724393, −2.13947982638133250048305426329, −1.46381003402249827491135508559, 0.870997882387445428837510785556, 3.03400981939804411592480648723, 4.08354703267216182881588452182, 5.40653066857275769862985971062, 5.63748434339351426619926897148, 6.64039416734445678874498579728, 7.79807229881362798826964523406, 8.709603667862278307507657022537, 9.321507346758037792206171380227, 10.12706448809514413212314355383

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