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Properties

Label 2-2808-2808.2131-c0-0-3
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  + (0.173 − 0.984i)2-s + (0.396 + 0.918i)3-s + (−0.939 − 0.342i)4-s + (0.914 − 0.767i)5-s + (0.973 − 0.230i)6-s + (1.28 − 0.469i)7-s + (−0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.597 − 1.03i)10-s + (−0.0581 − 0.998i)12-s + (0.173 + 0.984i)13-s + (−0.238 − 1.35i)14-s + (1.06 + 0.536i)15-s + (0.766 + 0.642i)16-s + (0.0581 + 0.100i)17-s + (0.597 + 0.802i)18-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.396 + 0.918i)3-s + (−0.939 − 0.342i)4-s + (0.914 − 0.767i)5-s + (0.973 − 0.230i)6-s + (1.28 − 0.469i)7-s + (−0.5 + 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.597 − 1.03i)10-s + (−0.0581 − 0.998i)12-s + (0.173 + 0.984i)13-s + (−0.238 − 1.35i)14-s + (1.06 + 0.536i)15-s + (0.766 + 0.642i)16-s + (0.0581 + 0.100i)17-s + (0.597 + 0.802i)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: $\chi_{2808} (2131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :0),\ 0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.825857060\)
\(L(\frac12)\) \(\approx\) \(1.825857060\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (-0.396 - 0.918i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
good5 \( 1 + (-0.914 + 0.767i)T + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (-1.28 + 0.469i)T + (0.766 - 0.642i)T^{2} \)
11 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.0581 - 0.100i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
37 \( 1 + (0.396 + 0.686i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (1.28 + 1.07i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (1.67 - 0.611i)T + (0.766 - 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.973 + 1.68i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−8.910935039285608928834945867212, −8.596634520094021665197877263403, −7.78020787689475078240170134482, −6.33316350107904107948399295264, −5.28678819318462626550863106652, −4.81036649029846072142815178724, −4.27745098227709173819875350144, −3.27040453187956708447987347238, −2.05460985410890433052592389575, −1.44311081822586615772929545087, 1.35244072588745663038281270053, 2.51398278303194796388763223185, 3.28521315346639390880303006254, 4.69378604620188534418952529782, 5.45374112133114262748748200500, 6.18766237813184188767139173082, 6.68884344967584355254136441284, 7.61868574531903674121008501325, 8.252462190131080734397593763258, 8.566699577560788834921905976857

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