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Properties

Label 2-2848-712.555-c0-0-0
Degree $2$
Conductor $2848$
Sign $0.660 + 0.750i$
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  + (1.56 − 0.340i)3-s + (1.42 − 0.649i)9-s + (−0.989 − 1.14i)11-s + (0.540 − 1.84i)17-s + (0.0498 − 0.133i)19-s + (0.142 + 0.989i)25-s + (0.722 − 0.540i)27-s + (−1.93 − 1.45i)33-s + (−0.300 + 1.38i)41-s + (0.956 + 0.0683i)43-s + (−0.989 + 0.142i)49-s + (0.219 − 3.06i)51-s + (0.0325 − 0.226i)57-s + (1.71 + 0.373i)59-s + (1.53 + 0.983i)67-s + ⋯
L(s)  = 1  + (1.56 − 0.340i)3-s + (1.42 − 0.649i)9-s + (−0.989 − 1.14i)11-s + (0.540 − 1.84i)17-s + (0.0498 − 0.133i)19-s + (0.142 + 0.989i)25-s + (0.722 − 0.540i)27-s + (−1.93 − 1.45i)33-s + (−0.300 + 1.38i)41-s + (0.956 + 0.0683i)43-s + (−0.989 + 0.142i)49-s + (0.219 − 3.06i)51-s + (0.0325 − 0.226i)57-s + (1.71 + 0.373i)59-s + (1.53 + 0.983i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2848\)    =    \(2^{5} \cdot 89\)
Sign: $0.660 + 0.750i$
Analytic conductor: \(1.42133\)
Root analytic conductor: \(1.19219\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2848} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2848,\ (\ :0),\ 0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.020796954\)
\(L(\frac12)\) \(\approx\) \(2.020796954\)
\(L(1)\) 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 \( 1 + (-1.56 + 0.340i)T + (0.909 - 0.415i)T^{2} \)
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 \( 1 + (-0.0498 + 0.133i)T + (-0.755 - 0.654i)T^{2} \)
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 \( 1 + (0.300 - 1.38i)T + (-0.909 - 0.415i)T^{2} \)
43 \( 1 + (-0.956 - 0.0683i)T + (0.989 + 0.142i)T^{2} \)
47 \( 1 + (0.415 - 0.909i)T^{2} \)
53 \( 1 + (0.415 + 0.909i)T^{2} \)
59 \( 1 + (-1.71 - 0.373i)T + (0.909 + 0.415i)T^{2} \)
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 \( 1 + (1.71 - 0.936i)T + (0.540 - 0.841i)T^{2} \)
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

−8.636502653246678064913926332216, −8.259893659871464498774353231495, −7.44114647910637765103146246660, −6.99386717676817742687147688045, −5.72022185542772517259983681706, −4.99809497372693428432039045419, −3.79566704665807015250297143605, −2.91449161436202310617136670954, −2.61424578709930083670253654773, −1.14194712890650319760724082272, 1.82431735558588485543622168467, 2.44165240402509103054659129562, 3.48265214426300312241933813693, 4.12542244249886668659722310021, 4.99238380539043016048687992242, 6.02779314389138755265742982503, 7.09080323609829490492773026918, 7.82839508546719599710960955045, 8.287054421728755428244731645949, 8.929729298697833924776328177515

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