Properties

Label 2-536-536.435-c0-0-0
Degree $2$
Conductor $536$
Sign $0.685 - 0.727i$
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  + (0.0475 + 0.998i)2-s + (1.50 − 0.442i)3-s + (−0.995 + 0.0950i)4-s + (0.514 + 1.48i)6-s + (−0.142 − 0.989i)8-s + (1.23 − 0.795i)9-s + (0.0930 − 0.268i)11-s + (−1.45 + 0.584i)12-s + (0.981 − 0.189i)16-s + (−1.15 − 0.110i)17-s + (0.853 + 1.19i)18-s + (−1.74 + 0.899i)19-s + (0.273 + 0.0801i)22-s + (−0.653 − 1.43i)24-s + (−0.142 + 0.989i)25-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)2-s + (1.50 − 0.442i)3-s + (−0.995 + 0.0950i)4-s + (0.514 + 1.48i)6-s + (−0.142 − 0.989i)8-s + (1.23 − 0.795i)9-s + (0.0930 − 0.268i)11-s + (−1.45 + 0.584i)12-s + (0.981 − 0.189i)16-s + (−1.15 − 0.110i)17-s + (0.853 + 1.19i)18-s + (−1.74 + 0.899i)19-s + (0.273 + 0.0801i)22-s + (−0.653 − 1.43i)24-s + (−0.142 + 0.989i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $0.685 - 0.727i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ 0.685 - 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246629414\)
\(L(\frac12)\) \(\approx\) \(1.246629414\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0475 - 0.998i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
good3 \( 1 + (-1.50 + 0.442i)T + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (0.142 - 0.989i)T^{2} \)
7 \( 1 + (-0.580 - 0.814i)T^{2} \)
11 \( 1 + (-0.0930 + 0.268i)T + (-0.786 - 0.618i)T^{2} \)
13 \( 1 + (-0.235 - 0.971i)T^{2} \)
17 \( 1 + (1.15 + 0.110i)T + (0.981 + 0.189i)T^{2} \)
19 \( 1 + (1.74 - 0.899i)T + (0.580 - 0.814i)T^{2} \)
23 \( 1 + (0.888 + 0.458i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.235 + 0.971i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.975 + 1.37i)T + (-0.327 - 0.945i)T^{2} \)
43 \( 1 + (0.271 + 0.595i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + (-0.0475 - 0.998i)T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.264 + 1.83i)T + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (0.786 - 0.618i)T^{2} \)
71 \( 1 + (-0.981 + 0.189i)T^{2} \)
73 \( 1 + (-0.651 - 1.88i)T + (-0.786 + 0.618i)T^{2} \)
79 \( 1 + (-0.723 + 0.690i)T^{2} \)
83 \( 1 + (1.28 - 0.247i)T + (0.928 - 0.371i)T^{2} \)
89 \( 1 + (0.0913 + 0.0268i)T + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (-0.995 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
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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

−11.01282271909130652820004300465, −9.853767924435592699006593495310, −8.877121475221182621244684216523, −8.558990329620347339534781507375, −7.62388643859935597021185150689, −6.85939645007146186788041314809, −5.87272733978529516498529031658, −4.38357172718491100256091532316, −3.51877897408444117894329912764, −2.06538464953225577092620536470, 2.10281290068202996215980681425, 2.81525805335228881672199546626, 4.13535074264435909667938881766, 4.58228298823544602731557090694, 6.37059521103547196146704955349, 7.79035125214781339254220288253, 8.727710517442326891649202626098, 9.036856589761190851032401412429, 10.06323513059919787522288241628, 10.71852648214121148609895814497

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