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Properties

Label 2-2848-712.187-c0-0-0
Degree $2$
Conductor $2848$
Sign $0.987 - 0.155i$
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  + (1.94 + 0.139i)3-s + (2.79 + 0.401i)9-s + (0.540 − 0.158i)11-s + (−0.755 − 0.345i)17-s + (−1.40 + 1.05i)19-s + (−0.841 − 0.540i)25-s + (3.47 + 0.755i)27-s + (1.07 − 0.234i)33-s + (−0.100 − 1.41i)41-s + (−0.898 + 1.64i)43-s + (0.540 − 0.841i)49-s + (−1.42 − 0.778i)51-s + (−2.88 + 1.85i)57-s + (−0.697 + 0.0498i)59-s + (−1.29 − 1.49i)67-s + ⋯
L(s)  = 1  + (1.94 + 0.139i)3-s + (2.79 + 0.401i)9-s + (0.540 − 0.158i)11-s + (−0.755 − 0.345i)17-s + (−1.40 + 1.05i)19-s + (−0.841 − 0.540i)25-s + (3.47 + 0.755i)27-s + (1.07 − 0.234i)33-s + (−0.100 − 1.41i)41-s + (−0.898 + 1.64i)43-s + (0.540 − 0.841i)49-s + (−1.42 − 0.778i)51-s + (−2.88 + 1.85i)57-s + (−0.697 + 0.0498i)59-s + (−1.29 − 1.49i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2848\)    =    \(2^{5} \cdot 89\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(1.42133\)
Root analytic conductor: \(1.19219\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2848} (1967, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2848,\ (\ :0),\ 0.987 - 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.435743064\)
\(L(\frac12)\) \(\approx\) \(2.435743064\)
\(L(1)\) 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 \( 1 + (0.841 - 0.540i)T \)
good3 \( 1 + (-1.94 - 0.139i)T + (0.989 + 0.142i)T^{2} \)
5 \( 1 + (0.841 + 0.540i)T^{2} \)
7 \( 1 + (-0.540 + 0.841i)T^{2} \)
11 \( 1 + (-0.540 + 0.158i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.989 + 0.142i)T^{2} \)
17 \( 1 + (0.755 + 0.345i)T + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (1.40 - 1.05i)T + (0.281 - 0.959i)T^{2} \)
23 \( 1 + (0.281 - 0.959i)T^{2} \)
29 \( 1 + (0.540 - 0.841i)T^{2} \)
31 \( 1 + (0.281 + 0.959i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.100 + 1.41i)T + (-0.989 + 0.142i)T^{2} \)
43 \( 1 + (0.898 - 1.64i)T + (-0.540 - 0.841i)T^{2} \)
47 \( 1 + (-0.142 - 0.989i)T^{2} \)
53 \( 1 + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.697 - 0.0498i)T + (0.989 - 0.142i)T^{2} \)
61 \( 1 + (0.909 + 0.415i)T^{2} \)
67 \( 1 + (1.29 + 1.49i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (0.841 - 0.540i)T^{2} \)
73 \( 1 + (0.153 + 1.07i)T + (-0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.697 - 1.86i)T + (-0.755 + 0.654i)T^{2} \)
97 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
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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

−8.911727722496887773792645389060, −8.298942365574828729969115411452, −7.77518173037504471007887354694, −6.88099353975935452656767898865, −6.16425160712603952211240568847, −4.70390606638526649963019591582, −4.03790179720182564889034353072, −3.39281243138490040077956051953, −2.34642565522031264132162675877, −1.70415883632960076188836487206, 1.60814379354100433054641155303, 2.33900189361644404275308884137, 3.21637091904746636460832029360, 4.11888455415419377889910720060, 4.60478185915220211922773999481, 6.15796216587761969482599613659, 6.96953366371597787559364517109, 7.48248740201727536451625090185, 8.539610948834755187597908428234, 8.691652578860705855111307159301

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