Properties

Label 2-2808-936.571-c0-0-3
Degree $2$
Conductor $2808$
Sign $0.866 - 0.5i$
Analytic cond. $1.40137$
Root an. cond. $1.18379$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.766 − 1.32i)5-s + (−0.173 − 0.300i)7-s − 0.999·8-s + 1.53·10-s + (−0.5 + 0.866i)13-s + (0.173 − 0.300i)14-s + (−0.5 − 0.866i)16-s + 1.87·17-s + (0.766 + 1.32i)20-s + (−0.673 − 1.16i)25-s − 0.999·26-s + 0.347·28-s + (0.5 − 0.866i)31-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.766 − 1.32i)5-s + (−0.173 − 0.300i)7-s − 0.999·8-s + 1.53·10-s + (−0.5 + 0.866i)13-s + (0.173 − 0.300i)14-s + (−0.5 − 0.866i)16-s + 1.87·17-s + (0.766 + 1.32i)20-s + (−0.673 − 1.16i)25-s − 0.999·26-s + 0.347·28-s + (0.5 − 0.866i)31-s + (0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(1.40137\)
Root analytic conductor: \(1.18379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (1819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :0),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.699459007\)
\(L(\frac12)\) \(\approx\) \(1.699459007\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.87T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.53T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 0.347T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.021073997213878036094775919606, −8.106899369410880587548614385003, −7.63625073741670834978923810334, −6.61186611111138613464968857565, −5.87828508162788075394050923481, −5.24045570336551185413641941502, −4.55237577424930014543191498964, −3.76700983914554452879312746881, −2.52771322891463967329343798997, −1.11340207452450151835295177409, 1.32895126404561611978888288606, 2.75557608694147840339422236317, 2.83908497302343051494416634656, 3.91216246553446994489744036426, 5.16119840390072679833862102174, 5.76602233670483120960436197330, 6.35107774991621668797622593955, 7.34240479716333537416330302585, 8.149093408070892003222491639418, 9.421853164877690230736000855764

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