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Properties

Label 2-2888-2888.251-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.388 + 0.921i$
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  + (0.0642 + 0.997i)2-s + (0.890 − 1.68i)3-s + (−0.991 + 0.128i)4-s + (1.73 + 0.780i)6-s + (−0.191 − 0.981i)8-s + (−1.47 − 2.16i)9-s + (1.62 − 0.458i)11-s + (−0.667 + 1.78i)12-s + (0.967 − 0.254i)16-s + (−0.632 − 1.51i)17-s + (2.06 − 1.61i)18-s + (−0.896 + 0.443i)19-s + (0.561 + 1.58i)22-s + (−1.82 − 0.551i)24-s + (−0.00918 + 0.999i)25-s + ⋯
L(s)  = 1  + (0.0642 + 0.997i)2-s + (0.890 − 1.68i)3-s + (−0.991 + 0.128i)4-s + (1.73 + 0.780i)6-s + (−0.191 − 0.981i)8-s + (−1.47 − 2.16i)9-s + (1.62 − 0.458i)11-s + (−0.667 + 1.78i)12-s + (0.967 − 0.254i)16-s + (−0.632 − 1.51i)17-s + (2.06 − 1.61i)18-s + (−0.896 + 0.443i)19-s + (0.561 + 1.58i)22-s + (−1.82 − 0.551i)24-s + (−0.00918 + 0.999i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.388 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.480238978\)
\(L(\frac12)\) \(\approx\) \(1.480238978\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0642 - 0.997i)T \)
19 \( 1 + (0.896 - 0.443i)T \)
good3 \( 1 + (-0.890 + 1.68i)T + (-0.562 - 0.826i)T^{2} \)
5 \( 1 + (0.00918 - 0.999i)T^{2} \)
7 \( 1 + (-0.904 + 0.426i)T^{2} \)
11 \( 1 + (-1.62 + 0.458i)T + (0.851 - 0.523i)T^{2} \)
13 \( 1 + (-0.100 + 0.994i)T^{2} \)
17 \( 1 + (0.632 + 1.51i)T + (-0.703 + 0.710i)T^{2} \)
23 \( 1 + (-0.997 - 0.0734i)T^{2} \)
29 \( 1 + (-0.577 - 0.816i)T^{2} \)
31 \( 1 + (0.298 - 0.954i)T^{2} \)
37 \( 1 + (0.879 + 0.475i)T^{2} \)
41 \( 1 + (-0.385 - 0.164i)T + (0.690 + 0.723i)T^{2} \)
43 \( 1 + (1.94 - 0.398i)T + (0.919 - 0.393i)T^{2} \)
47 \( 1 + (0.979 - 0.200i)T^{2} \)
53 \( 1 + (-0.315 + 0.948i)T^{2} \)
59 \( 1 + (-1.55 + 1.16i)T + (0.280 - 0.959i)T^{2} \)
61 \( 1 + (-0.741 - 0.670i)T^{2} \)
67 \( 1 + (-0.869 + 0.0159i)T + (0.999 - 0.0367i)T^{2} \)
71 \( 1 + (-0.741 + 0.670i)T^{2} \)
73 \( 1 + (-1.11 + 1.46i)T + (-0.263 - 0.964i)T^{2} \)
79 \( 1 + (0.800 + 0.599i)T^{2} \)
83 \( 1 + (0.948 + 0.657i)T + (0.350 + 0.936i)T^{2} \)
89 \( 1 + (-1.09 - 1.43i)T + (-0.263 + 0.964i)T^{2} \)
97 \( 1 + (1.53 + 0.0281i)T + (0.999 + 0.0367i)T^{2} \)
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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

−8.606763404013110547350761940861, −8.006482711102522590317982619824, −7.14531811759791719569959442805, −6.70083112857917861842157787657, −6.24816634087882606510060674956, −5.19398647462761586408705591153, −3.92469894648156508763472741651, −3.22683686539483131973547044423, −1.97690660561337787114117540262, −0.853287977283225026614862184807, 1.82988820834354658035583897392, 2.63501026304314864044748043420, 3.83656410442785464716456987220, 4.02433995643272280035909511391, 4.68430140863300150721250989565, 5.71672756063676515851169337029, 6.75211942058965325773294875960, 8.345472523948866571218304093256, 8.532045779976130451699626074226, 9.208834221021909266444791900904

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