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Properties

Label 2-1564-1564.407-c0-0-1
Degree $2$
Conductor $1564$
Sign $-0.988 - 0.153i$
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  + (−0.654 + 0.755i)2-s + (−1.61 + 1.03i)3-s + (−0.142 − 0.989i)4-s + (0.273 − 1.89i)6-s + (0.273 − 0.0801i)7-s + (0.841 + 0.540i)8-s + (1.11 − 2.44i)9-s + (−1.10 − 1.27i)11-s + (1.25 + 1.45i)12-s + (0.273 + 0.0801i)13-s + (−0.118 + 0.258i)14-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)17-s + (1.11 + 2.44i)18-s + (−0.357 + 0.412i)21-s + 1.68·22-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (−1.61 + 1.03i)3-s + (−0.142 − 0.989i)4-s + (0.273 − 1.89i)6-s + (0.273 − 0.0801i)7-s + (0.841 + 0.540i)8-s + (1.11 − 2.44i)9-s + (−1.10 − 1.27i)11-s + (1.25 + 1.45i)12-s + (0.273 + 0.0801i)13-s + (−0.118 + 0.258i)14-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)17-s + (1.11 + 2.44i)18-s + (−0.357 + 0.412i)21-s + 1.68·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1564\)    =    \(2^{2} \cdot 17 \cdot 23\)
Sign: $-0.988 - 0.153i$
Analytic conductor: \(0.780537\)
Root analytic conductor: \(0.883480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1564} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1564,\ (\ :0),\ -0.988 - 0.153i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2852529280\)
\(L(\frac12)\) \(\approx\) \(0.2852529280\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
good3 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
11 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
13 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
19 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.959 + 0.281i)T^{2} \)
31 \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \)
37 \( 1 + (0.654 + 0.755i)T^{2} \)
41 \( 1 + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (-0.415 + 0.909i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.415 - 0.909i)T^{2} \)
67 \( 1 + (0.142 + 0.989i)T^{2} \)
71 \( 1 + (1.30 - 1.51i)T + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (-1.41 + 0.909i)T + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (0.654 - 0.755i)T^{2} \)
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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

−10.11704647742389681271747741313, −9.332758470824117069569497072200, −8.440367125762274710314108735486, −7.62995780683371352389283441435, −6.44961365035030134812025315130, −5.94032718528012477294636099521, −5.31056542548161227544396789686, −4.58144492030959011245665797689, −3.46421990344886334718301617217, −1.22645797292648512642937965728, 0.38917297766133889846721287478, 1.81680292317503947328616934802, 2.57263859370714147047767560083, 4.55033696774225110187159753679, 4.91985378909898504846441457494, 6.18182662831400110533501801254, 6.90821471190708056366165339627, 7.77299386841021567417300623185, 8.100064847728385727579103663417, 9.579042018878561679922741392093

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