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Properties

Label 2-2848-712.339-c0-0-0
Degree $2$
Conductor $2848$
Sign $0.970 - 0.240i$
Analytic cond. $1.42133$
Root an. cond. $1.19219$
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  + (−0.254 + 1.17i)3-s + (−0.397 − 0.181i)9-s + (0.989 − 1.14i)11-s + (−0.540 − 1.84i)17-s + (1.86 − 0.697i)19-s + (0.142 − 0.989i)25-s + (−0.404 + 0.540i)27-s + (1.08 + 1.45i)33-s + (−1.38 + 0.300i)41-s + (−0.125 − 1.75i)43-s + (0.989 + 0.142i)49-s + (2.29 − 0.164i)51-s + (0.340 + 2.36i)57-s + (0.203 + 0.936i)59-s + (−1.53 + 0.983i)67-s + ⋯
L(s)  = 1  + (−0.254 + 1.17i)3-s + (−0.397 − 0.181i)9-s + (0.989 − 1.14i)11-s + (−0.540 − 1.84i)17-s + (1.86 − 0.697i)19-s + (0.142 − 0.989i)25-s + (−0.404 + 0.540i)27-s + (1.08 + 1.45i)33-s + (−1.38 + 0.300i)41-s + (−0.125 − 1.75i)43-s + (0.989 + 0.142i)49-s + (2.29 − 0.164i)51-s + (0.340 + 2.36i)57-s + (0.203 + 0.936i)59-s + (−1.53 + 0.983i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2848\)    =    \(2^{5} \cdot 89\)
Sign: $0.970 - 0.240i$
Analytic conductor: \(1.42133\)
Root analytic conductor: \(1.19219\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2848} (2831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2848,\ (\ :0),\ 0.970 - 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220606150\)
\(L(\frac12)\) \(\approx\) \(1.220606150\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 + (-0.142 - 0.989i)T \)
good3 \( 1 + (0.254 - 1.17i)T + (-0.909 - 0.415i)T^{2} \)
5 \( 1 + (-0.142 + 0.989i)T^{2} \)
7 \( 1 + (-0.989 - 0.142i)T^{2} \)
11 \( 1 + (-0.989 + 1.14i)T + (-0.142 - 0.989i)T^{2} \)
13 \( 1 + (-0.909 - 0.415i)T^{2} \)
17 \( 1 + (0.540 + 1.84i)T + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (-1.86 + 0.697i)T + (0.755 - 0.654i)T^{2} \)
23 \( 1 + (0.755 - 0.654i)T^{2} \)
29 \( 1 + (0.989 + 0.142i)T^{2} \)
31 \( 1 + (0.755 + 0.654i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (1.38 - 0.300i)T + (0.909 - 0.415i)T^{2} \)
43 \( 1 + (0.125 + 1.75i)T + (-0.989 + 0.142i)T^{2} \)
47 \( 1 + (0.415 + 0.909i)T^{2} \)
53 \( 1 + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (-0.203 - 0.936i)T + (-0.909 + 0.415i)T^{2} \)
61 \( 1 + (-0.281 - 0.959i)T^{2} \)
67 \( 1 + (1.53 - 0.983i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (-0.822 - 1.80i)T + (-0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.654 + 0.755i)T^{2} \)
83 \( 1 + (0.203 - 0.373i)T + (-0.540 - 0.841i)T^{2} \)
97 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
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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

−9.078577532785331112850915205338, −8.590460028333662892776783533755, −7.31757529211775527385404051351, −6.80691891417104513988342357787, −5.64390716455997450377031214898, −5.12937625316327803766132251114, −4.30372405546213477846736373870, −3.48259436585740392922433699006, −2.69008702894295899106183789954, −0.917455454645741597968775067618, 1.44321702589581572114159503778, 1.77595083720488894456201023072, 3.31169721272842477362905671073, 4.15889977201855654093332617889, 5.19751719514624662020998749270, 6.16502912107087099334634356357, 6.63742101853134102047145693420, 7.45555878175323341541981509457, 7.88705829975268021166423879928, 8.912267470268019251323904365165

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