Properties

Label 2-1557-519.518-c0-0-2
Degree $2$
Conductor $1557$
Sign $0.577 + 0.816i$
Analytic cond. $0.777044$
Root an. cond. $0.881501$
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  − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·14-s − 16-s − 1.41i·19-s − 22-s + 1.41i·29-s − 31-s − 1.41i·35-s + 37-s + 1.41i·38-s + ⋯
L(s)  = 1  − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·14-s − 16-s − 1.41i·19-s − 22-s + 1.41i·29-s − 31-s − 1.41i·35-s + 37-s + 1.41i·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1557 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1557\)    =    \(3^{2} \cdot 173\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(0.777044\)
Root analytic conductor: \(0.881501\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1557} (1556, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1557,\ (\ :0),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7522793412\)
\(L(\frac12)\) \(\approx\) \(0.7522793412\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
173 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.41iT - T^{2} \)
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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

−9.565835919330122940495317411516, −8.918049519468224049440511336496, −8.062079165240402230773928756239, −6.95829102912759421221076918846, −6.79875023021112684648148062836, −5.36843104611759082607566228157, −4.48356734584018720913082804307, −3.55341216909524798486923754681, −1.93390413920784301225517886575, −0.947627843752204265299253020835, 1.54176698445586549480915237533, 2.27535401652704931647262932643, 3.76128663308943664194000900046, 4.93519528013636861941628711067, 5.92069177501044245728103029382, 6.34444248279772160107631369976, 7.70225793479821420465170857462, 8.324959752860125911620926140612, 9.230048199855122154226755921513, 9.490355893158025642147342753944

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