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Properties

Label 2-2888-152.123-c0-0-6
Degree $2$
Conductor $2888$
Sign $-0.963 + 0.268i$
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  + (−0.173 − 0.984i)2-s + (1.43 − 1.20i)3-s + (−0.939 + 0.342i)4-s + (−1.43 − 1.20i)6-s + (0.5 + 0.866i)8-s + (0.439 − 2.49i)9-s + (−0.173 − 0.300i)11-s + (−0.939 + 1.62i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 2.53·18-s + (−0.266 + 0.223i)22-s + (1.76 + 0.642i)24-s + (0.766 + 0.642i)25-s + (−1.43 − 2.49i)27-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (1.43 − 1.20i)3-s + (−0.939 + 0.342i)4-s + (−1.43 − 1.20i)6-s + (0.5 + 0.866i)8-s + (0.439 − 2.49i)9-s + (−0.173 − 0.300i)11-s + (−0.939 + 1.62i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 2.53·18-s + (−0.266 + 0.223i)22-s + (1.76 + 0.642i)24-s + (0.766 + 0.642i)25-s + (−1.43 − 2.49i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.963 + 0.268i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2555, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.963 + 0.268i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.629268622\)
\(L(\frac12)\) \(\approx\) \(1.629268622\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 \)
good3 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
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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

−8.555608019496060020471390248031, −8.129803766920396192568361265079, −7.30819494257483086639253993206, −6.69453303093352819793273760349, −5.43510584024851808860799093701, −4.33540087818991102298854061720, −3.28862571742974189249721026601, −2.83941151805954343301759843271, −1.93464264334566206416743839366, −0.970248826454434833075147303003, 1.89164294543425103488461011377, 3.08001907118846920551201387259, 3.94332968671675169952241773074, 4.55722562736011883808942773113, 5.28295290101834581203899157779, 6.31188555785500791207606034798, 7.31697704291472777775999784862, 7.959278780336978161958398866557, 8.624399042877374225477463195445, 9.038961735234814246415133343761

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