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Properties

Label 2-536-536.131-c0-0-0
Degree $2$
Conductor $536$
Sign $-0.355 + 0.934i$
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  + (−0.959 − 0.281i)2-s + (0.186 − 1.29i)3-s + (0.841 + 0.540i)4-s + (−0.544 + 1.19i)6-s + (−0.654 − 0.755i)8-s + (−0.686 − 0.201i)9-s + (−0.544 − 1.19i)11-s + (0.857 − 0.989i)12-s + (0.415 + 0.909i)16-s + (1.41 − 0.909i)17-s + (0.601 + 0.386i)18-s + (−0.797 + 0.234i)19-s + (0.186 + 1.29i)22-s + (−1.10 + 0.708i)24-s + (−0.654 + 0.755i)25-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (0.186 − 1.29i)3-s + (0.841 + 0.540i)4-s + (−0.544 + 1.19i)6-s + (−0.654 − 0.755i)8-s + (−0.686 − 0.201i)9-s + (−0.544 − 1.19i)11-s + (0.857 − 0.989i)12-s + (0.415 + 0.909i)16-s + (1.41 − 0.909i)17-s + (0.601 + 0.386i)18-s + (−0.797 + 0.234i)19-s + (0.186 + 1.29i)22-s + (−1.10 + 0.708i)24-s + (−0.654 + 0.755i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $-0.355 + 0.934i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ -0.355 + 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6120424280\)
\(L(\frac12)\) \(\approx\) \(0.6120424280\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
good3 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.841 - 0.540i)T^{2} \)
11 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.142 + 0.989i)T^{2} \)
17 \( 1 + (-1.41 + 0.909i)T + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
23 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.142 - 0.989i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (0.959 + 0.281i)T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (-0.857 - 0.989i)T + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
79 \( 1 + (0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \)
89 \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 - 1.68T + T^{2} \)
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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

−10.75794920122154250406158360026, −9.879449633015542252828234513462, −8.801125478589872052768686701268, −8.004623398074930340035289876978, −7.47989703216868265987233885331, −6.50655161710064429552020008774, −5.56330107488505992307934075152, −3.46510149826820625476157469668, −2.41407759098035826291519141369, −1.06229202303170221751699867361, 2.10136173346203005041419062098, 3.59836608190874179622730583879, 4.79158869582494977773254662450, 5.76436370550255850839347950191, 6.98356573792304216932049091853, 7.982999545030760122777882752287, 8.753638796599623803865798952590, 9.815441941915953373522298340506, 10.13574315682816934658387782067, 10.77552629905251736733054538500

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