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Properties

Label 2-2808-936.571-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.462529992\)
\(L(\frac12)\) \(\approx\) \(1.462529992\)
\(L(1)\) 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

−9.085812541616724578802810923427, −8.537055507355489232360596573804, −7.83023019419428698884554723494, −6.89130693128968879056178215398, −6.18893531916722373551521728474, −5.37922119449085374338935418450, −4.82688927277148582615362743975, −4.11736787987896011511390806745, −2.69058569325040982868417568263, −1.98950187653924777123905140316, 0.78353535400114509445401010587, 1.95686677742981913589531432341, 2.93528081555984216211104886028, 3.95956553388492202963162497915, 4.62159580546650659645236845382, 5.19615836990663366663523103668, 6.44288255574021230085117444632, 7.00319040898116722973061118355, 8.012951078040588256828808906516, 8.660853207658247528499599151627

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