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Properties

Label 2-2888-2888.43-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.357 - 0.933i$
Analytic cond. $1.44129$
Root an. cond. $1.20054$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.888 + 0.459i)2-s + (1.02 + 0.461i)3-s + (0.577 + 0.816i)4-s + (0.698 + 0.880i)6-s + (0.137 + 0.990i)8-s + (0.175 + 0.197i)9-s + (0.0532 − 1.93i)11-s + (0.215 + 1.10i)12-s + (−0.333 + 0.942i)16-s + (0.532 + 1.15i)17-s + (0.0649 + 0.256i)18-s + (−0.0459 + 0.998i)19-s + (0.936 − 1.69i)22-s + (−0.315 + 1.07i)24-s + (−0.155 + 0.987i)25-s + ⋯
L(s)  = 1  + (0.888 + 0.459i)2-s + (1.02 + 0.461i)3-s + (0.577 + 0.816i)4-s + (0.698 + 0.880i)6-s + (0.137 + 0.990i)8-s + (0.175 + 0.197i)9-s + (0.0532 − 1.93i)11-s + (0.215 + 1.10i)12-s + (−0.333 + 0.942i)16-s + (0.532 + 1.15i)17-s + (0.0649 + 0.256i)18-s + (−0.0459 + 0.998i)19-s + (0.936 − 1.69i)22-s + (−0.315 + 1.07i)24-s + (−0.155 + 0.987i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.357 - 0.933i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.357 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.888695183\)
\(L(\frac12)\) \(\approx\) \(2.888695183\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.888 - 0.459i)T \)
19 \( 1 + (0.0459 - 0.998i)T \)
good3 \( 1 + (-1.02 - 0.461i)T + (0.663 + 0.748i)T^{2} \)
5 \( 1 + (0.155 - 0.987i)T^{2} \)
7 \( 1 + (-0.350 + 0.936i)T^{2} \)
11 \( 1 + (-0.0532 + 1.93i)T + (-0.998 - 0.0550i)T^{2} \)
13 \( 1 + (-0.989 - 0.146i)T^{2} \)
17 \( 1 + (-0.532 - 1.15i)T + (-0.649 + 0.760i)T^{2} \)
23 \( 1 + (-0.315 - 0.948i)T^{2} \)
29 \( 1 + (0.861 + 0.507i)T^{2} \)
31 \( 1 + (-0.904 - 0.426i)T^{2} \)
37 \( 1 + (-0.546 + 0.837i)T^{2} \)
41 \( 1 + (0.724 + 0.482i)T + (0.384 + 0.922i)T^{2} \)
43 \( 1 + (0.571 - 0.172i)T + (0.832 - 0.554i)T^{2} \)
47 \( 1 + (-0.957 + 0.289i)T^{2} \)
53 \( 1 + (0.729 + 0.684i)T^{2} \)
59 \( 1 + (-0.0780 + 1.21i)T + (-0.991 - 0.128i)T^{2} \)
61 \( 1 + (-0.997 + 0.0734i)T^{2} \)
67 \( 1 + (-1.86 + 0.602i)T + (0.811 - 0.584i)T^{2} \)
71 \( 1 + (-0.997 - 0.0734i)T^{2} \)
73 \( 1 + (1.65 + 0.152i)T + (0.983 + 0.182i)T^{2} \)
79 \( 1 + (-0.0642 - 0.997i)T^{2} \)
83 \( 1 + (1.09 + 1.32i)T + (-0.191 + 0.981i)T^{2} \)
89 \( 1 + (0.698 - 0.0643i)T + (0.983 - 0.182i)T^{2} \)
97 \( 1 + (1.45 + 0.470i)T + (0.811 + 0.584i)T^{2} \)
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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

−8.663673209737787608006118955587, −8.387659791224688451012985532269, −7.79734953818444313588932747609, −6.68310922228279197916868585698, −5.86806814066015279348653662123, −5.41792420663364781662712317468, −4.09158065169987639218305511391, −3.48192681234661458822558187659, −3.11478126775997575785578116727, −1.79548838373010256724580925325, 1.45374340709191789541923623024, 2.45407571991073628135309037472, 2.84168175843840366661802782435, 4.07371703244104252279728543759, 4.75541664586460823407439374957, 5.50111500308920908493286100902, 6.83026800394072036483682070709, 7.10130324715007788933897267740, 7.898891493618785018656163993651, 8.897027592106310172036025949286

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