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Properties

Label 2-1557-519.518-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.7964915119\)
\(L(\frac12)\) \(\approx\) \(0.7964915119\)
\(L(1)\) 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

−9.882100179514305725913547209887, −8.922280636046533671087612600388, −8.345523152203410392106412646763, −7.58596541082838204627691527792, −6.41356676929251892677435884004, −5.46370903452346113168792723288, −5.14493191129841691554501260915, −3.93420004713447177983923367226, −3.27685611069167612332900559098, −2.22448097382365254428889267730, 0.43944716434828770620498518772, 2.64865689306291155330659417459, 3.63276283596606777575182113673, 4.30843548309651575955451710462, 4.88426456119280864751110425569, 5.94528301480961380371396082373, 7.00076166694260413787766874734, 7.62529435469117780657530363194, 8.355766052940446791035915645614, 9.422022201807035411463585512867

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