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Properties

Label 2-2888-2888.2851-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.603 + 0.797i$
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  + (−0.401 + 0.915i)2-s + (1.24 − 0.969i)3-s + (−0.677 − 0.735i)4-s + (0.387 + 1.52i)6-s + (0.945 − 0.324i)8-s + (0.366 − 1.44i)9-s + (−0.332 − 1.31i)11-s + (−1.55 − 0.259i)12-s + (−0.0825 + 0.996i)16-s + (0.111 − 0.121i)17-s + (1.17 + 0.915i)18-s + (−0.401 − 0.915i)19-s + (1.33 + 0.222i)22-s + (0.863 − 1.32i)24-s + (−0.986 − 0.164i)25-s + ⋯
L(s)  = 1  + (−0.401 + 0.915i)2-s + (1.24 − 0.969i)3-s + (−0.677 − 0.735i)4-s + (0.387 + 1.52i)6-s + (0.945 − 0.324i)8-s + (0.366 − 1.44i)9-s + (−0.332 − 1.31i)11-s + (−1.55 − 0.259i)12-s + (−0.0825 + 0.996i)16-s + (0.111 − 0.121i)17-s + (1.17 + 0.915i)18-s + (−0.401 − 0.915i)19-s + (1.33 + 0.222i)22-s + (0.863 − 1.32i)24-s + (−0.986 − 0.164i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.603 + 0.797i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.603 + 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.330600292\)
\(L(\frac12)\) \(\approx\) \(1.330600292\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.401 - 0.915i)T \)
19 \( 1 + (0.401 + 0.915i)T \)
good3 \( 1 + (-1.24 + 0.969i)T + (0.245 - 0.969i)T^{2} \)
5 \( 1 + (0.986 + 0.164i)T^{2} \)
7 \( 1 + (0.0825 + 0.996i)T^{2} \)
11 \( 1 + (0.332 + 1.31i)T + (-0.879 + 0.475i)T^{2} \)
13 \( 1 + (-0.245 + 0.969i)T^{2} \)
17 \( 1 + (-0.111 + 0.121i)T + (-0.0825 - 0.996i)T^{2} \)
23 \( 1 + (-0.245 - 0.969i)T^{2} \)
29 \( 1 + (0.0825 + 0.996i)T^{2} \)
31 \( 1 + (0.677 - 0.735i)T^{2} \)
37 \( 1 + (0.879 - 0.475i)T^{2} \)
41 \( 1 + (-0.863 - 1.32i)T + (-0.401 + 0.915i)T^{2} \)
43 \( 1 + (-0.706 + 0.382i)T + (0.546 - 0.837i)T^{2} \)
47 \( 1 + (0.879 - 0.475i)T^{2} \)
53 \( 1 + (0.879 - 0.475i)T^{2} \)
59 \( 1 + (0.439 + 0.672i)T + (-0.401 + 0.915i)T^{2} \)
61 \( 1 + (-0.789 - 0.614i)T^{2} \)
67 \( 1 + (-0.464 + 0.159i)T + (0.789 - 0.614i)T^{2} \)
71 \( 1 + (-0.789 + 0.614i)T^{2} \)
73 \( 1 + (0.740 - 0.804i)T + (-0.0825 - 0.996i)T^{2} \)
79 \( 1 + (-0.546 + 0.837i)T^{2} \)
83 \( 1 + (-0.0136 - 0.164i)T + (-0.986 + 0.164i)T^{2} \)
89 \( 1 + (-0.111 - 0.121i)T + (-0.0825 + 0.996i)T^{2} \)
97 \( 1 + (-1.89 - 0.649i)T + (0.789 + 0.614i)T^{2} \)
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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

−8.689173852659170273352423293914, −7.981149825388349669089245509968, −7.60504699870060569150520649544, −6.69808887883444741725453891916, −6.11967092486809230124258672474, −5.19901006702062558878079486258, −4.04592714290303567233211864080, −3.06384393698599154149375393944, −2.09225500776940051549266986524, −0.834612886834476268783758918502, 1.79211958601181526944688340774, 2.47396804815464588696785142982, 3.40962708184052238991263621063, 4.16881523939286399991231668811, 4.63469101955808965501807302262, 5.80254686149085691427669269519, 7.34582312898264550892514209722, 7.79314046290039468407480612220, 8.544053089532189065793438562905, 9.273479171577715983472252860468

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