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Properties

Label 2-1564-1564.271-c0-0-0
Degree $2$
Conductor $1564$
Sign $-0.672 - 0.740i$
Analytic cond. $0.780537$
Root an. cond. $0.883480$
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  + (0.959 + 0.281i)2-s + (−1.19 + 1.37i)3-s + (0.841 + 0.540i)4-s + (−1.53 + 0.983i)6-s + (0.449 + 0.983i)7-s + (0.654 + 0.755i)8-s + (−0.328 − 2.28i)9-s + (1.45 − 0.425i)11-s + (−1.74 + 0.512i)12-s + (−0.698 + 1.53i)13-s + (0.153 + 1.07i)14-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + (0.328 − 2.28i)18-s + (−1.88 − 0.554i)21-s + 1.51·22-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)2-s + (−1.19 + 1.37i)3-s + (0.841 + 0.540i)4-s + (−1.53 + 0.983i)6-s + (0.449 + 0.983i)7-s + (0.654 + 0.755i)8-s + (−0.328 − 2.28i)9-s + (1.45 − 0.425i)11-s + (−1.74 + 0.512i)12-s + (−0.698 + 1.53i)13-s + (0.153 + 1.07i)14-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + (0.328 − 2.28i)18-s + (−1.88 − 0.554i)21-s + 1.51·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1564\)    =    \(2^{2} \cdot 17 \cdot 23\)
Sign: $-0.672 - 0.740i$
Analytic conductor: \(0.780537\)
Root analytic conductor: \(0.883480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1564} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1564,\ (\ :0),\ -0.672 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.489263050\)
\(L(\frac12)\) \(\approx\) \(1.489263050\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 - 0.281i)T \)
17 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (0.540 + 0.841i)T \)
good3 \( 1 + (1.19 - 1.37i)T + (-0.142 - 0.989i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
7 \( 1 + (-0.449 - 0.983i)T + (-0.654 + 0.755i)T^{2} \)
11 \( 1 + (-1.45 + 0.425i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.698 - 1.53i)T + (-0.654 - 0.755i)T^{2} \)
19 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 + (-0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.989 + 1.14i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (0.959 - 0.281i)T^{2} \)
41 \( 1 + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (0.142 + 0.989i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (-0.841 - 0.540i)T^{2} \)
71 \( 1 + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (-0.234 + 0.512i)T + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (0.857 - 0.989i)T + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (0.959 + 0.281i)T^{2} \)
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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

−9.850355150383671871801809290376, −9.353753497235926102496030726281, −8.460662462070233436848922059573, −7.11657413138598084114907424761, −6.23020973567013089492922353039, −5.78087705658568076219453258855, −4.90653130117297799742488181401, −4.25299725202695622838188567391, −3.58003442089418497861639487735, −2.06036815788449116676039737733, 1.12831812391049122556205172135, 1.79820596423359557664866495602, 3.38693096231286037971040754715, 4.42519868920478823044725057817, 5.45319070652119968645415406758, 5.89277156039099519975860917892, 6.88551539744296053234041930438, 7.42546650506925530534821052722, 7.950116607539297869544281050635, 9.719083900161847986471715458529

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