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Properties

Label 2-2808-2808.571-c0-0-2
Degree $2$
Conductor $2808$
Sign $0.727 - 0.686i$
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  + (−0.939 − 0.342i)2-s + (0.973 + 0.230i)3-s + (0.766 + 0.642i)4-s + (−0.238 + 1.35i)5-s + (−0.835 − 0.549i)6-s + (1.36 − 1.14i)7-s + (−0.500 − 0.866i)8-s + (0.893 + 0.448i)9-s + (0.686 − 1.18i)10-s + (0.597 + 0.802i)12-s + (−0.939 + 0.342i)13-s + (−1.67 + 0.611i)14-s + (−0.543 + 1.26i)15-s + (0.173 + 0.984i)16-s + (−0.597 + 1.03i)17-s + (−0.686 − 0.727i)18-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.973 + 0.230i)3-s + (0.766 + 0.642i)4-s + (−0.238 + 1.35i)5-s + (−0.835 − 0.549i)6-s + (1.36 − 1.14i)7-s + (−0.500 − 0.866i)8-s + (0.893 + 0.448i)9-s + (0.686 − 1.18i)10-s + (0.597 + 0.802i)12-s + (−0.939 + 0.342i)13-s + (−1.67 + 0.611i)14-s + (−0.543 + 1.26i)15-s + (0.173 + 0.984i)16-s + (−0.597 + 1.03i)17-s + (−0.686 − 0.727i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.727 - 0.686i$
Analytic conductor: \(1.40137\)
Root analytic conductor: \(1.18379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :0),\ 0.727 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.238457780\)
\(L(\frac12)\) \(\approx\) \(1.238457780\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (-0.973 - 0.230i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
good5 \( 1 + (0.238 - 1.35i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (-1.36 + 1.14i)T + (0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.597 - 1.03i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.973 - 1.68i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.344 + 1.95i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.606 + 0.509i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.835 + 1.44i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.766 - 0.642i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T^{2} \)
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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

−8.930908076734534882609630209771, −8.253988796548122583961004018188, −7.75783514653278140462977859283, −6.94914749032827507118798764772, −6.74699776428626310173123198382, −4.88217323342117933667771639520, −4.01753228867325357319580793070, −3.31090039881288723596383449851, −2.32283540282767621134263854230, −1.56809388649595537732228152707, 1.02770620646769842385205222993, 2.09718839417670034042011964697, 2.68017770848595359157709092476, 4.45541067990638157864979108417, 4.99755917416948584553293888075, 5.77890833232572161090731435533, 6.99258633657968223395152791800, 7.86163331113314279617134207729, 8.114877891180861143915730077643, 8.870018971638551566668958356516

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