Properties

Label 2-536-536.531-c0-0-0
Degree $2$
Conductor $536$
Sign $-0.798 - 0.602i$
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  + (−0.654 + 0.755i)2-s + (0.698 + 1.53i)3-s + (−0.142 − 0.989i)4-s + (−1.61 − 0.474i)6-s + (0.841 + 0.540i)8-s + (−1.19 + 1.38i)9-s + (−1.61 + 0.474i)11-s + (1.41 − 0.909i)12-s + (−0.959 + 0.281i)16-s + (0.0405 − 0.281i)17-s + (−0.260 − 1.81i)18-s + (1.25 + 1.45i)19-s + (0.698 − 1.53i)22-s + (−0.239 + 1.66i)24-s + (0.841 − 0.540i)25-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (0.698 + 1.53i)3-s + (−0.142 − 0.989i)4-s + (−1.61 − 0.474i)6-s + (0.841 + 0.540i)8-s + (−1.19 + 1.38i)9-s + (−1.61 + 0.474i)11-s + (1.41 − 0.909i)12-s + (−0.959 + 0.281i)16-s + (0.0405 − 0.281i)17-s + (−0.260 − 1.81i)18-s + (1.25 + 1.45i)19-s + (0.698 − 1.53i)22-s + (−0.239 + 1.66i)24-s + (0.841 − 0.540i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $-0.798 - 0.602i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ -0.798 - 0.602i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7360446957\)
\(L(\frac12)\) \(\approx\) \(0.7360446957\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 - 0.755i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
good3 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
5 \( 1 + (-0.841 + 0.540i)T^{2} \)
7 \( 1 + (0.142 + 0.989i)T^{2} \)
11 \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (-0.415 + 0.909i)T^{2} \)
17 \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \)
19 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
23 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.415 - 0.909i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.654 - 0.755i)T^{2} \)
53 \( 1 + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \)
61 \( 1 + (-0.841 - 0.540i)T^{2} \)
71 \( 1 + (0.959 - 0.281i)T^{2} \)
73 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (-0.415 + 0.909i)T^{2} \)
83 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + 0.284T + T^{2} \)
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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

−10.64799543252047697629602903193, −10.33455272038624175126956215063, −9.655961054913502501190805377440, −8.767053031231123981472762061164, −8.028608109590587805216572591386, −7.21231233638758520413216725506, −5.52484798039744859195897799776, −5.08282791755478151274583780175, −3.84427770870222337577330776376, −2.47055069026038840425819143713, 1.11566733756708459161912107450, 2.60452187156297139317554090367, 3.09018930093223900587762207983, 5.00960081716739590539091804079, 6.52002486212915778765982053823, 7.54084027094721985404506548697, 7.922530497650540089478569684273, 8.818329787515971685407546547040, 9.654886033671284524077877546089, 10.88151632055067000890875081289

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