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Properties

Label 2-2848-712.587-c0-0-0
Degree $2$
Conductor $2848$
Sign $0.769 - 0.639i$
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  + (0.898 − 0.334i)3-s + (−0.0614 + 0.0532i)9-s + (0.281 + 1.95i)11-s + (0.909 + 1.41i)17-s + (−1.59 − 0.114i)19-s + (0.959 + 0.281i)25-s + (−0.496 + 0.909i)27-s + (0.909 + 1.66i)33-s + (0.494 − 1.32i)41-s + (0.254 − 0.340i)43-s + (0.281 − 0.959i)49-s + (1.29 + 0.966i)51-s + (−1.47 + 0.432i)57-s + (−1.83 − 0.682i)59-s + (0.627 − 1.37i)67-s + ⋯
L(s)  = 1  + (0.898 − 0.334i)3-s + (−0.0614 + 0.0532i)9-s + (0.281 + 1.95i)11-s + (0.909 + 1.41i)17-s + (−1.59 − 0.114i)19-s + (0.959 + 0.281i)25-s + (−0.496 + 0.909i)27-s + (0.909 + 1.66i)33-s + (0.494 − 1.32i)41-s + (0.254 − 0.340i)43-s + (0.281 − 0.959i)49-s + (1.29 + 0.966i)51-s + (−1.47 + 0.432i)57-s + (−1.83 − 0.682i)59-s + (0.627 − 1.37i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2848\)    =    \(2^{5} \cdot 89\)
Sign: $0.769 - 0.639i$
Analytic conductor: \(1.42133\)
Root analytic conductor: \(1.19219\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2848} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2848,\ (\ :0),\ 0.769 - 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.588154460\)
\(L(\frac12)\) \(\approx\) \(1.588154460\)
\(L(1)\) 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 \( 1 + (-0.959 + 0.281i)T \)
good3 \( 1 + (-0.898 + 0.334i)T + (0.755 - 0.654i)T^{2} \)
5 \( 1 + (-0.959 - 0.281i)T^{2} \)
7 \( 1 + (-0.281 + 0.959i)T^{2} \)
11 \( 1 + (-0.281 - 1.95i)T + (-0.959 + 0.281i)T^{2} \)
13 \( 1 + (0.755 - 0.654i)T^{2} \)
17 \( 1 + (-0.909 - 1.41i)T + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (1.59 + 0.114i)T + (0.989 + 0.142i)T^{2} \)
23 \( 1 + (0.989 + 0.142i)T^{2} \)
29 \( 1 + (0.281 - 0.959i)T^{2} \)
31 \( 1 + (0.989 - 0.142i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.494 + 1.32i)T + (-0.755 - 0.654i)T^{2} \)
43 \( 1 + (-0.254 + 0.340i)T + (-0.281 - 0.959i)T^{2} \)
47 \( 1 + (-0.654 + 0.755i)T^{2} \)
53 \( 1 + (-0.654 - 0.755i)T^{2} \)
59 \( 1 + (1.83 + 0.682i)T + (0.755 + 0.654i)T^{2} \)
61 \( 1 + (-0.540 - 0.841i)T^{2} \)
67 \( 1 + (-0.627 + 1.37i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (0.368 - 0.425i)T + (-0.142 - 0.989i)T^{2} \)
79 \( 1 + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (-1.83 - 0.398i)T + (0.909 + 0.415i)T^{2} \)
97 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
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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

−8.937050998645752851404070910668, −8.303729571924890591165285736359, −7.59764010217330661346577371440, −6.96487164301041599265386882287, −6.14359296218537227502930058752, −5.07058317572930441062235248240, −4.23455324644460373670408533222, −3.42116187892835871096403556893, −2.21066374281361135118134517163, −1.75136513063545855611787163365, 0.928295354707605664080491586797, 2.63365035072473211080681606025, 3.10732494952240647396893101813, 3.95154217822887410021043122173, 4.90298382761699259272991517306, 5.98121607832702126060866824761, 6.43349696619934864123793101830, 7.67399157180537469123431229294, 8.230637055867146364103143097530, 8.984957286200284674531481844792

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