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Properties

Label 2-2848-712.427-c0-0-0
Degree $2$
Conductor $2848$
Sign $-0.0526 - 0.998i$
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  + (−1.12 + 1.50i)3-s + (−0.707 − 2.40i)9-s + (0.909 + 0.584i)11-s + (0.989 + 0.857i)17-s + (0.936 − 1.71i)19-s + (−0.415 + 0.909i)25-s + (2.65 + 0.989i)27-s + (−1.89 + 0.708i)33-s + (1.13 − 0.847i)41-s + (0.0303 + 0.139i)43-s + (0.909 + 0.415i)49-s + (−2.39 + 0.521i)51-s + (1.52 + 3.33i)57-s + (1.19 + 1.59i)59-s + (0.0801 + 0.557i)67-s + ⋯
L(s)  = 1  + (−1.12 + 1.50i)3-s + (−0.707 − 2.40i)9-s + (0.909 + 0.584i)11-s + (0.989 + 0.857i)17-s + (0.936 − 1.71i)19-s + (−0.415 + 0.909i)25-s + (2.65 + 0.989i)27-s + (−1.89 + 0.708i)33-s + (1.13 − 0.847i)41-s + (0.0303 + 0.139i)43-s + (0.909 + 0.415i)49-s + (−2.39 + 0.521i)51-s + (1.52 + 3.33i)57-s + (1.19 + 1.59i)59-s + (0.0801 + 0.557i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0526 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0526 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2848\)    =    \(2^{5} \cdot 89\)
Sign: $-0.0526 - 0.998i$
Analytic conductor: \(1.42133\)
Root analytic conductor: \(1.19219\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2848} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2848,\ (\ :0),\ -0.0526 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8929563590\)
\(L(\frac12)\) \(\approx\) \(0.8929563590\)
\(L(1)\) 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 \( 1 + (0.415 + 0.909i)T \)
good3 \( 1 + (1.12 - 1.50i)T + (-0.281 - 0.959i)T^{2} \)
5 \( 1 + (0.415 - 0.909i)T^{2} \)
7 \( 1 + (-0.909 - 0.415i)T^{2} \)
11 \( 1 + (-0.909 - 0.584i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (-0.281 - 0.959i)T^{2} \)
17 \( 1 + (-0.989 - 0.857i)T + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (-0.936 + 1.71i)T + (-0.540 - 0.841i)T^{2} \)
23 \( 1 + (-0.540 - 0.841i)T^{2} \)
29 \( 1 + (0.909 + 0.415i)T^{2} \)
31 \( 1 + (-0.540 + 0.841i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-1.13 + 0.847i)T + (0.281 - 0.959i)T^{2} \)
43 \( 1 + (-0.0303 - 0.139i)T + (-0.909 + 0.415i)T^{2} \)
47 \( 1 + (-0.959 - 0.281i)T^{2} \)
53 \( 1 + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (-1.19 - 1.59i)T + (-0.281 + 0.959i)T^{2} \)
61 \( 1 + (0.755 + 0.654i)T^{2} \)
67 \( 1 + (-0.0801 - 0.557i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (1.74 + 0.512i)T + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (1.19 - 0.0855i)T + (0.989 - 0.142i)T^{2} \)
97 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
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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

−9.250146545214344676175084333786, −8.875604205512393064510745956035, −7.46416757766840546068936080601, −6.77801157476410474601208481027, −5.75804559469401451096236875297, −5.40532204156501445441783479473, −4.38739692820916818504643437762, −3.93082929858039954589039835293, −2.93505946077012033567502164735, −1.11707929944069021128331711202, 0.859517914070966052977480549078, 1.66269026588261661085720065991, 2.89622869981921211039823598126, 4.07189873639668645952574701413, 5.34213390607023887113206990505, 5.78494027196507675797724571734, 6.44604208260434943624476146337, 7.21768385613500084244426589543, 7.84207351366268368035754943163, 8.434646954833962367286853587398

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