Properties

Label 2-1566-9.4-c1-0-22
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: $\chi_{1566} (523, \cdot )$
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

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

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