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Properties

Label 2-536-536.523-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  + (−0.786 + 0.618i)2-s + (1.30 − 1.50i)3-s + (0.235 − 0.971i)4-s + (−0.0947 + 1.98i)6-s + (0.415 + 0.909i)8-s + (−0.421 − 2.93i)9-s + (0.0395 + 0.829i)11-s + (−1.15 − 1.62i)12-s + (−0.888 − 0.458i)16-s + (0.341 + 1.40i)17-s + (2.14 + 2.04i)18-s + (−1.65 + 0.660i)19-s + (−0.544 − 0.627i)22-s + (1.91 + 0.560i)24-s + (0.415 − 0.909i)25-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (1.30 − 1.50i)3-s + (0.235 − 0.971i)4-s + (−0.0947 + 1.98i)6-s + (0.415 + 0.909i)8-s + (−0.421 − 2.93i)9-s + (0.0395 + 0.829i)11-s + (−1.15 − 1.62i)12-s + (−0.888 − 0.458i)16-s + (0.341 + 1.40i)17-s + (2.14 + 2.04i)18-s + (−1.65 + 0.660i)19-s + (−0.544 − 0.627i)22-s + (1.91 + 0.560i)24-s + (0.415 − 0.909i)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: $\chi_{536} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ 0.685 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9004061941\)
\(L(\frac12)\) \(\approx\) \(0.9004061941\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.786 - 0.618i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
good3 \( 1 + (-1.30 + 1.50i)T + (-0.142 - 0.989i)T^{2} \)
5 \( 1 + (-0.415 + 0.909i)T^{2} \)
7 \( 1 + (-0.723 - 0.690i)T^{2} \)
11 \( 1 + (-0.0395 - 0.829i)T + (-0.995 + 0.0950i)T^{2} \)
13 \( 1 + (-0.981 + 0.189i)T^{2} \)
17 \( 1 + (-0.341 - 1.40i)T + (-0.888 + 0.458i)T^{2} \)
19 \( 1 + (1.65 - 0.660i)T + (0.723 - 0.690i)T^{2} \)
23 \( 1 + (-0.928 - 0.371i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.981 - 0.189i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.205 - 0.196i)T + (0.0475 - 0.998i)T^{2} \)
43 \( 1 + (0.0913 + 0.0268i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (0.786 - 0.618i)T^{2} \)
53 \( 1 + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (-0.481 - 1.05i)T + (-0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.995 + 0.0950i)T^{2} \)
71 \( 1 + (0.888 + 0.458i)T^{2} \)
73 \( 1 + (-0.0224 + 0.470i)T + (-0.995 - 0.0950i)T^{2} \)
79 \( 1 + (0.327 - 0.945i)T^{2} \)
83 \( 1 + (1.49 + 0.770i)T + (0.580 + 0.814i)T^{2} \)
89 \( 1 + (-1.02 - 1.18i)T + (-0.142 + 0.989i)T^{2} \)
97 \( 1 + (0.235 + 0.408i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.63104772139669101209272699066, −9.748869215902211961889744234771, −8.700722124663867871772705360560, −8.284504423975887242778046749490, −7.50748373645156132685562600005, −6.62740543144828090128175964350, −6.03782706348742245783982177709, −4.07946887140238529610165137421, −2.41811566403519489068148741322, −1.54038099917372231128586946478, 2.35190038931418778041758521939, 3.19788332236590255257398207654, 4.10492377791198289369691758494, 5.16723423859116912890593972468, 7.05465984621885280082764998354, 8.140921245179870707532291147636, 8.766513342326841116045035703537, 9.343020823271195108777863156219, 10.13944950178345303131616488764, 10.90191539563143026361753613611

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