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Properties

Label 2-2888-2888.2547-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.767 + 0.641i$
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  + (0.546 − 0.837i)2-s + (1.78 + 0.614i)3-s + (−0.401 − 0.915i)4-s + (1.49 − 1.16i)6-s + (−0.986 − 0.164i)8-s + (2.03 + 1.58i)9-s + (−0.633 + 0.493i)11-s + (−0.156 − 1.88i)12-s + (−0.677 + 0.735i)16-s + (0.544 − 1.24i)17-s + (2.43 − 0.837i)18-s + (0.546 + 0.837i)19-s + (0.0663 + 0.800i)22-s + (−1.66 − 0.900i)24-s + (−0.0825 − 0.996i)25-s + ⋯
L(s)  = 1  + (0.546 − 0.837i)2-s + (1.78 + 0.614i)3-s + (−0.401 − 0.915i)4-s + (1.49 − 1.16i)6-s + (−0.986 − 0.164i)8-s + (2.03 + 1.58i)9-s + (−0.633 + 0.493i)11-s + (−0.156 − 1.88i)12-s + (−0.677 + 0.735i)16-s + (0.544 − 1.24i)17-s + (2.43 − 0.837i)18-s + (0.546 + 0.837i)19-s + (0.0663 + 0.800i)22-s + (−1.66 − 0.900i)24-s + (−0.0825 − 0.996i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.767 + 0.641i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.767 + 0.641i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.797319327\)
\(L(\frac12)\) \(\approx\) \(2.797319327\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.546 + 0.837i)T \)
19 \( 1 + (-0.546 - 0.837i)T \)
good3 \( 1 + (-1.78 - 0.614i)T + (0.789 + 0.614i)T^{2} \)
5 \( 1 + (0.0825 + 0.996i)T^{2} \)
7 \( 1 + (0.677 + 0.735i)T^{2} \)
11 \( 1 + (0.633 - 0.493i)T + (0.245 - 0.969i)T^{2} \)
13 \( 1 + (-0.789 - 0.614i)T^{2} \)
17 \( 1 + (-0.544 + 1.24i)T + (-0.677 - 0.735i)T^{2} \)
23 \( 1 + (-0.789 + 0.614i)T^{2} \)
29 \( 1 + (0.677 + 0.735i)T^{2} \)
31 \( 1 + (0.401 - 0.915i)T^{2} \)
37 \( 1 + (-0.245 + 0.969i)T^{2} \)
41 \( 1 + (1.66 - 0.900i)T + (0.546 - 0.837i)T^{2} \)
43 \( 1 + (-0.268 + 1.06i)T + (-0.879 - 0.475i)T^{2} \)
47 \( 1 + (-0.245 + 0.969i)T^{2} \)
53 \( 1 + (-0.245 + 0.969i)T^{2} \)
59 \( 1 + (0.962 - 0.520i)T + (0.546 - 0.837i)T^{2} \)
61 \( 1 + (-0.945 + 0.324i)T^{2} \)
67 \( 1 + (1.55 + 0.259i)T + (0.945 + 0.324i)T^{2} \)
71 \( 1 + (-0.945 - 0.324i)T^{2} \)
73 \( 1 + (-0.706 + 1.61i)T + (-0.677 - 0.735i)T^{2} \)
79 \( 1 + (0.879 + 0.475i)T^{2} \)
83 \( 1 + (-0.917 - 0.996i)T + (-0.0825 + 0.996i)T^{2} \)
89 \( 1 + (-0.544 - 1.24i)T + (-0.677 + 0.735i)T^{2} \)
97 \( 1 + (1.97 - 0.329i)T + (0.945 - 0.324i)T^{2} \)
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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

−9.141574278099733868382603551991, −8.204558534535448396297316808990, −7.68955904174194010805455194960, −6.67861038737140133413224428536, −5.29581669736297059949336646577, −4.75754546982507356200869200231, −3.86982946289667180662260445029, −3.15153846130347060246137498505, −2.52680374141296196778568321259, −1.61521376664178128278675387352, 1.59852539364749974710546586165, 2.85471033847065705244197290732, 3.31041590397182962478139590129, 4.14344943765772759342663443961, 5.20126861177500219441605104977, 6.17460419426404791191583790321, 6.97515763289087999434223005775, 7.65665622171670036153896198789, 8.103107706501436152617584371543, 8.769802681316258117675574649636

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