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Properties

Label 2-2808-936.571-c0-0-4
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

\(L(\frac{1}{2})\) \(\approx\) \(1.204035277\)
\(L(\frac12)\) \(\approx\) \(1.204035277\)
\(L(1)\) 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

−8.903752336629302013567777021062, −8.342969270809709638368015401594, −7.84332871866147519794988801386, −6.29746029831466161670154670024, −5.36394941776667482894442825797, −5.04442692406078595259839326779, −4.05975863225977907507757771955, −2.70186698595627849511834931484, −1.92498346421381928771151318593, −1.07325261089522034517326325899, 1.38790115458303616844080222582, 2.35897544538286635196880520181, 3.77772152122551983612911582404, 4.51382251616826448514028692641, 5.64391704629935696748568838365, 6.40511565010206596097423324158, 6.86291927484631352399562698582, 7.53989442245495021620232363036, 8.148112961470725348404338958688, 9.370645454892460579837966981411

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