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Properties

Label 2-536-536.283-c0-0-0
Degree $2$
Conductor $536$
Sign $0.567 + 0.823i$
Analytic cond. $0.267498$
Root an. cond. $0.517202$
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  + (0.841 − 0.540i)2-s + (0.273 − 0.0801i)3-s + (0.415 − 0.909i)4-s + (0.186 − 0.215i)6-s + (−0.142 − 0.989i)8-s + (−0.773 + 0.496i)9-s + (0.186 + 0.215i)11-s + (0.0405 − 0.281i)12-s + (−0.654 − 0.755i)16-s + (0.345 + 0.755i)17-s + (−0.381 + 0.835i)18-s + (−1.10 − 0.708i)19-s + (0.273 + 0.0801i)22-s + (−0.118 − 0.258i)24-s + (−0.142 + 0.989i)25-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (0.273 − 0.0801i)3-s + (0.415 − 0.909i)4-s + (0.186 − 0.215i)6-s + (−0.142 − 0.989i)8-s + (−0.773 + 0.496i)9-s + (0.186 + 0.215i)11-s + (0.0405 − 0.281i)12-s + (−0.654 − 0.755i)16-s + (0.345 + 0.755i)17-s + (−0.381 + 0.835i)18-s + (−1.10 − 0.708i)19-s + (0.273 + 0.0801i)22-s + (−0.118 − 0.258i)24-s + (−0.142 + 0.989i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $0.567 + 0.823i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ 0.567 + 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.419933883\)
\(L(\frac12)\) \(\approx\) \(1.419933883\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
good3 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (0.142 - 0.989i)T^{2} \)
7 \( 1 + (-0.415 + 0.909i)T^{2} \)
11 \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \)
19 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
23 \( 1 + (-0.841 + 0.540i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.959 - 0.281i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
43 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + (-0.841 + 0.540i)T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (0.142 + 0.989i)T^{2} \)
71 \( 1 + (0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
79 \( 1 + (0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.857 - 0.989i)T + (-0.142 + 0.989i)T^{2} \)
89 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
97 \( 1 - 0.830T + T^{2} \)
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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

−11.04404916434774925553334474187, −10.28816911605002672536029574754, −9.241807975541732032900823228409, −8.314744784916442390371012347952, −7.13290565853293873980547606714, −6.10846908822782626994896630217, −5.20793017014822518421317940359, −4.12673558727183440694705184682, −3.00402552163550373889347173199, −1.87483948587413780525913898953, 2.46488476542176465734140162535, 3.53823584278906208396379155504, 4.53067383950477896077546669105, 5.76564134817301501194477922011, 6.39019813066600564767597980776, 7.53919608603977546262288981665, 8.406040901864516028423712858885, 9.145426657644129013803161690426, 10.41170030215803698032761627789, 11.44708678299756049124592262987

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