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Properties

Label 2-2888-2888.2395-c0-0-0
Degree $2$
Conductor $2888$
Sign $-0.610 - 0.791i$
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  + (0.789 + 0.614i)2-s + (0.917 + 0.996i)3-s + (0.245 + 0.969i)4-s + (0.111 + 1.34i)6-s + (−0.401 + 0.915i)8-s + (−0.0689 + 0.831i)9-s + (−0.0405 − 0.489i)11-s + (−0.740 + 1.13i)12-s + (−0.879 + 0.475i)16-s + (−0.431 + 1.70i)17-s + (−0.565 + 0.614i)18-s + (0.789 − 0.614i)19-s + (0.268 − 0.411i)22-s + (−1.28 + 0.439i)24-s + (0.546 − 0.837i)25-s + ⋯
L(s)  = 1  + (0.789 + 0.614i)2-s + (0.917 + 0.996i)3-s + (0.245 + 0.969i)4-s + (0.111 + 1.34i)6-s + (−0.401 + 0.915i)8-s + (−0.0689 + 0.831i)9-s + (−0.0405 − 0.489i)11-s + (−0.740 + 1.13i)12-s + (−0.879 + 0.475i)16-s + (−0.431 + 1.70i)17-s + (−0.565 + 0.614i)18-s + (0.789 − 0.614i)19-s + (0.268 − 0.411i)22-s + (−1.28 + 0.439i)24-s + (0.546 − 0.837i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.610 - 0.791i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.610 - 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.535948141\)
\(L(\frac12)\) \(\approx\) \(2.535948141\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.789 - 0.614i)T \)
19 \( 1 + (-0.789 + 0.614i)T \)
good3 \( 1 + (-0.917 - 0.996i)T + (-0.0825 + 0.996i)T^{2} \)
5 \( 1 + (-0.546 + 0.837i)T^{2} \)
7 \( 1 + (0.879 + 0.475i)T^{2} \)
11 \( 1 + (0.0405 + 0.489i)T + (-0.986 + 0.164i)T^{2} \)
13 \( 1 + (0.0825 - 0.996i)T^{2} \)
17 \( 1 + (0.431 - 1.70i)T + (-0.879 - 0.475i)T^{2} \)
23 \( 1 + (0.0825 + 0.996i)T^{2} \)
29 \( 1 + (0.879 + 0.475i)T^{2} \)
31 \( 1 + (-0.245 + 0.969i)T^{2} \)
37 \( 1 + (0.986 - 0.164i)T^{2} \)
41 \( 1 + (1.28 + 0.439i)T + (0.789 + 0.614i)T^{2} \)
43 \( 1 + (1.55 - 0.259i)T + (0.945 - 0.324i)T^{2} \)
47 \( 1 + (0.986 - 0.164i)T^{2} \)
53 \( 1 + (0.986 - 0.164i)T^{2} \)
59 \( 1 + (-1.49 - 0.512i)T + (0.789 + 0.614i)T^{2} \)
61 \( 1 + (0.677 - 0.735i)T^{2} \)
67 \( 1 + (-0.0663 + 0.151i)T + (-0.677 - 0.735i)T^{2} \)
71 \( 1 + (0.677 + 0.735i)T^{2} \)
73 \( 1 + (-0.464 + 1.83i)T + (-0.879 - 0.475i)T^{2} \)
79 \( 1 + (-0.945 + 0.324i)T^{2} \)
83 \( 1 + (-1.54 - 0.837i)T + (0.546 + 0.837i)T^{2} \)
89 \( 1 + (0.431 + 1.70i)T + (-0.879 + 0.475i)T^{2} \)
97 \( 1 + (0.803 + 1.83i)T + (-0.677 + 0.735i)T^{2} \)
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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

−8.833748944019400483988391722357, −8.548704537448089361267932633022, −7.88332647797787775974156317780, −6.80182323441797997743834708672, −6.18470203319513651461345718984, −5.14666578644091214819256093834, −4.52612812734740344725526314469, −3.61812990872167559286530122125, −3.20653565547802850368478720599, −2.09466685346395851750847645247, 1.20725082709495101686716820283, 2.10965239290654361072887487662, 2.92624188952704255243143328853, 3.58663108321906401347888012831, 4.86017973560161820139224498653, 5.34082005338965174826650162508, 6.70190530639789418258857221982, 6.98102376205546506055249323280, 7.82810316972066887722677491523, 8.698896463631329590399961899443

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