Properties

Label 2-2816-88.43-c1-0-19
Degree $2$
Conductor $2816$
Sign $0.233 - 0.972i$
Analytic cond. $22.4858$
Root an. cond. $4.74192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·3-s + i·5-s + 2.82·7-s + (−1.73 − 2.82i)11-s − 4.89·13-s − 1.73i·15-s − 4.89i·17-s + 2.82i·19-s − 4.89·21-s + 5.19i·23-s + 4·25-s + 5.19·27-s + 1.73i·31-s + (2.99 + 4.89i)33-s + 2.82i·35-s + ⋯
L(s)  = 1  − 1.00·3-s + 0.447i·5-s + 1.06·7-s + (−0.522 − 0.852i)11-s − 1.35·13-s − 0.447i·15-s − 1.18i·17-s + 0.648i·19-s − 1.06·21-s + 1.08i·23-s + 0.800·25-s + 1.00·27-s + 0.311i·31-s + (0.522 + 0.852i)33-s + 0.478i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2816\)    =    \(2^{8} \cdot 11\)
Sign: $0.233 - 0.972i$
Analytic conductor: \(22.4858\)
Root analytic conductor: \(4.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2816} (1407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2816,\ (\ :1/2),\ 0.233 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8666785388\)
\(L(\frac12)\) \(\approx\) \(0.8666785388\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (1.73 + 2.82i)T \)
good3 \( 1 + 1.73T + 3T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 - 5.19iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + 3iT - 37T^{2} \)
41 \( 1 + 4.89iT - 41T^{2} \)
43 \( 1 - 5.65iT - 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 1.73T + 59T^{2} \)
61 \( 1 - 9.79T + 61T^{2} \)
67 \( 1 + 8.66T + 67T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−8.942885138285025997468770860767, −8.037707300708181733522242249925, −7.41762732866030629678921665253, −6.70246202998333478314323725902, −5.61083672875675637687483440900, −5.27055385801610600682808659349, −4.54607510790059365069932263929, −3.20843796761174913293009131326, −2.35644119655056678295309102120, −0.923929144017714830669273293190, 0.39716105346649715937193441987, 1.76680142225483998983371338343, 2.69680237253423014386167440856, 4.29816829769073521514887778730, 4.94796042622194755847176367855, 5.22242355315436964381918871228, 6.33894014520954805207871078370, 7.04388092696291133234525397169, 7.940574231588040275766340955635, 8.496537514396610144172250016295

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