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Properties

Label 2-2808-936.571-c0-0-0
Degree $2$
Conductor $2808$
Sign $i$
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 + (−0.939 + 1.62i)5-s + (−0.766 − 1.32i)7-s − 0.999·8-s − 1.87·10-s + (−0.5 + 0.866i)13-s + (0.766 − 1.32i)14-s + (−0.5 − 0.866i)16-s − 0.347·17-s + (−0.939 − 1.62i)20-s + (−1.26 − 2.19i)25-s − 0.999·26-s + 1.53·28-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 + (−0.939 + 1.62i)5-s + (−0.766 − 1.32i)7-s − 0.999·8-s − 1.87·10-s + (−0.5 + 0.866i)13-s + (0.766 − 1.32i)14-s + (−0.5 − 0.866i)16-s − 0.347·17-s + (−0.939 − 1.62i)20-s + (−1.26 − 2.19i)25-s − 0.999·26-s + 1.53·28-s + (0.5 − 0.866i)31-s + (0.499 − 0.866i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $i$
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: \((2,\ 2808,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1590276294\)
\(L(\frac12)\) \(\approx\) \(0.1590276294\)
\(L(1)\) 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 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + 0.347T + T^{2} \)
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 \( 1 + 1.87T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.53T + T^{2} \)
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

−9.566130954684943673803821538300, −8.514195822018399207417212228471, −7.58257936072070821162129865752, −7.16408062357919606169206962766, −6.72650651804821717012460362798, −6.09182308194470792868316506418, −4.69239170164381780224682960116, −3.94078217396504192312297011477, −3.47914214937857586765163882822, −2.54509936796200466928039440813, 0.083638204825075081030009234183, 1.51662194028972419632152534636, 2.74442327573442324846179389723, 3.52888488343768563498869120316, 4.48762252774925482148859025575, 5.24172518295086958813729517888, 5.62074298021922216581282972133, 6.75234654972324669827154868812, 7.989700196526752234122057149706, 8.729575540376612267965851479909

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