Properties

Label 2-1566-9.7-c1-0-3
Degree $2$
Conductor $1566$
Sign $0.424 - 0.905i$
Analytic cond. $12.5045$
Root an. cond. $3.53618$
Motivic weight $1$
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 + (−1.58 − 2.73i)5-s + (0.567 − 0.982i)7-s − 0.999·8-s − 3.16·10-s + (−2.89 + 5.01i)11-s + (2.82 + 4.89i)13-s + (−0.567 − 0.982i)14-s + (−0.5 + 0.866i)16-s − 2.86·17-s − 6.57·19-s + (−1.58 + 2.73i)20-s + (2.89 + 5.01i)22-s + (1.31 + 2.27i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.707 − 1.22i)5-s + (0.214 − 0.371i)7-s − 0.353·8-s − 1.00·10-s + (−0.873 + 1.51i)11-s + (0.784 + 1.35i)13-s + (−0.151 − 0.262i)14-s + (−0.125 + 0.216i)16-s − 0.694·17-s − 1.50·19-s + (−0.353 + 0.612i)20-s + (0.617 + 1.06i)22-s + (0.273 + 0.474i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1566\)    =    \(2 \cdot 3^{3} \cdot 29\)
Sign: $0.424 - 0.905i$
Analytic conductor: \(12.5045\)
Root analytic conductor: \(3.53618\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1566} (1045, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1566,\ (\ :1/2),\ 0.424 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5615168080\)
\(L(\frac12)\) \(\approx\) \(0.5615168080\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.58 + 2.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.567 + 0.982i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.89 - 5.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.82 - 4.89i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.86T + 17T^{2} \)
19 \( 1 + 6.57T + 19T^{2} \)
23 \( 1 + (-1.31 - 2.27i)T + (-11.5 + 19.9i)T^{2} \)
31 \( 1 + (3.00 + 5.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.114T + 37T^{2} \)
41 \( 1 + (-5.92 - 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.364 - 0.630i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.15 + 8.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.56T + 53T^{2} \)
59 \( 1 + (-3.05 - 5.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.82 - 4.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.86 + 4.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.36T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + (-5.62 + 9.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.99 - 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (3.15 - 5.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.456998688656447958788181063871, −8.936721942359892717897012516255, −8.086582100221065251426543661885, −7.25955793330189503214607576421, −6.26258774513308885435277375861, −5.04914028231501523829382385844, −4.29204296967152148996240893312, −4.11564061293095985230886649231, −2.34142252235321815954996491569, −1.43326311405634483784657814688, 0.19299882210203545846257586145, 2.59333051101579424321344590665, 3.26278806459063987231941261534, 4.13117985925485483550987857192, 5.41607459802227515993164609209, 6.03154195380973447417371429844, 6.80488904793053185825959018510, 7.71119522465545354084650630864, 8.380225905968566507643803904832, 8.830609501179799758484873659688

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