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Properties

Label 2-2808-936.571-c0-0-5
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 + (−0.173 + 0.300i)5-s + (−0.939 − 1.62i)7-s + 0.999·8-s + 0.347·10-s + (0.5 − 0.866i)13-s + (−0.939 + 1.62i)14-s + (−0.5 − 0.866i)16-s − 1.53·17-s + (−0.173 − 0.300i)20-s + (0.439 + 0.761i)25-s − 0.999·26-s + 1.87·28-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 + (−0.173 + 0.300i)5-s + (−0.939 − 1.62i)7-s + 0.999·8-s + 0.347·10-s + (0.5 − 0.866i)13-s + (−0.939 + 1.62i)14-s + (−0.5 − 0.866i)16-s − 1.53·17-s + (−0.173 − 0.300i)20-s + (0.439 + 0.761i)25-s − 0.999·26-s + 1.87·28-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

\(L(\frac{1}{2})\) \(\approx\) \(0.2955177241\)
\(L(\frac12)\) \(\approx\) \(0.2955177241\)
\(L(1)\) 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 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.53T + T^{2} \)
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 \( 1 + 0.347T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.87T + T^{2} \)
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

−8.703852412686794679430109483255, −7.85708496504730317910284436242, −6.97858545311660000805317344763, −6.72835364235516052785397827073, −5.25373462441020266188785531792, −4.19823612867088008797091594702, −3.58164931579764138223197677082, −2.92940487187282073148640491457, −1.53824782267641602597131885995, −0.22018328809072736702437035216, 1.77086727421514565803142752803, 2.78969335713640462938453510180, 4.16806957194241548386804165606, 4.90389680623335144345109766395, 5.91236759956200305954971752360, 6.35995654211189074691984786940, 6.96843080405474894570964737202, 8.179066817903947223103188875877, 8.651423546617661680343100883694, 9.306356430633043327638306913150

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