Properties

Label 2-2888-152.99-c0-0-4
Degree $2$
Conductor $2888$
Sign $0.600 + 0.799i$
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  + (−0.939 − 0.342i)2-s + (0.0603 + 0.342i)3-s + (0.766 + 0.642i)4-s + (0.0603 − 0.342i)6-s + (−0.500 − 0.866i)8-s + (0.826 − 0.300i)9-s + (−0.766 − 1.32i)11-s + (−0.173 + 0.300i)12-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 0.879·18-s + (0.266 + 1.50i)22-s + (0.266 − 0.223i)24-s + (0.173 − 0.984i)25-s + (0.326 + 0.565i)27-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.0603 + 0.342i)3-s + (0.766 + 0.642i)4-s + (0.0603 − 0.342i)6-s + (−0.500 − 0.866i)8-s + (0.826 − 0.300i)9-s + (−0.766 − 1.32i)11-s + (−0.173 + 0.300i)12-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 0.879·18-s + (0.266 + 1.50i)22-s + (0.266 − 0.223i)24-s + (0.173 − 0.984i)25-s + (0.326 + 0.565i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.600 + 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8247793220\)
\(L(\frac12)\) \(\approx\) \(0.8247793220\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 \)
good3 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
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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

−8.799054872368680127197708143403, −8.239386080958104371177213618742, −7.58722472711517702145188920425, −6.69127166966249938535329401587, −5.95178064962939379708802908074, −4.94948385914301494053131495363, −3.71840933796290334694385299841, −3.22138785840418679001733084434, −2.04537014333292216988041936923, −0.77396925366645034722253824972, 1.30600540388488115218042404049, 2.12490429876190549764824483364, 3.18720549978135750266237000919, 4.67374699823887096215235922960, 5.22727253203601877809763216298, 6.31145828428025367824746050584, 7.06770753850614674119761719468, 7.64366226451302693605959154531, 8.027053944660760286216294428816, 9.135323148476310038430720655334

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