Properties

Label 2-536-536.155-c0-0-0
Degree $2$
Conductor $536$
Sign $0.454 - 0.890i$
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  + (0.981 − 0.189i)2-s + (−0.738 + 1.61i)3-s + (0.928 − 0.371i)4-s + (−0.419 + 1.72i)6-s + (0.841 − 0.540i)8-s + (−1.41 − 1.63i)9-s + (0.396 + 1.63i)11-s + (−0.0845 + 1.77i)12-s + (0.723 − 0.690i)16-s + (−1.45 − 0.584i)17-s + (−1.69 − 1.33i)18-s + (−0.473 − 1.36i)19-s + (0.698 + 1.53i)22-s + (0.252 + 1.75i)24-s + (0.841 + 0.540i)25-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)2-s + (−0.738 + 1.61i)3-s + (0.928 − 0.371i)4-s + (−0.419 + 1.72i)6-s + (0.841 − 0.540i)8-s + (−1.41 − 1.63i)9-s + (0.396 + 1.63i)11-s + (−0.0845 + 1.77i)12-s + (0.723 − 0.690i)16-s + (−1.45 − 0.584i)17-s + (−1.69 − 1.33i)18-s + (−0.473 − 1.36i)19-s + (0.698 + 1.53i)22-s + (0.252 + 1.75i)24-s + (0.841 + 0.540i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $0.454 - 0.890i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ 0.454 - 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.249934363\)
\(L(\frac12)\) \(\approx\) \(1.249934363\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.981 + 0.189i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
good3 \( 1 + (0.738 - 1.61i)T + (-0.654 - 0.755i)T^{2} \)
5 \( 1 + (-0.841 - 0.540i)T^{2} \)
7 \( 1 + (0.786 + 0.618i)T^{2} \)
11 \( 1 + (-0.396 - 1.63i)T + (-0.888 + 0.458i)T^{2} \)
13 \( 1 + (-0.580 + 0.814i)T^{2} \)
17 \( 1 + (1.45 + 0.584i)T + (0.723 + 0.690i)T^{2} \)
19 \( 1 + (0.473 + 1.36i)T + (-0.786 + 0.618i)T^{2} \)
23 \( 1 + (0.327 - 0.945i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.580 - 0.814i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.02 + 0.809i)T + (0.235 - 0.971i)T^{2} \)
43 \( 1 + (0.0671 + 0.466i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.981 + 0.189i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.0800 + 0.0514i)T + (0.415 - 0.909i)T^{2} \)
61 \( 1 + (0.888 + 0.458i)T^{2} \)
71 \( 1 + (-0.723 + 0.690i)T^{2} \)
73 \( 1 + (-0.437 + 1.80i)T + (-0.888 - 0.458i)T^{2} \)
79 \( 1 + (0.995 + 0.0950i)T^{2} \)
83 \( 1 + (1.38 - 1.32i)T + (0.0475 - 0.998i)T^{2} \)
89 \( 1 + (-0.815 - 1.78i)T + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
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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

−11.07922149878781202204762459132, −10.64745336522849978817160238266, −9.607620078217792636576464374608, −9.050770807495384995925705645596, −7.11446429895121615543777434349, −6.44312090483078982774684987663, −5.08727093125525148719460057576, −4.68281371216619557712654891305, −3.89662600556954586727333919958, −2.47084844526413006591070800537, 1.53947678530118681467843482374, 2.86375611488258122159026035960, 4.33402561726446339729311037965, 5.77999220609451254213893816738, 6.18236824919423385704008202367, 6.88723198044754964805787191818, 8.012834273645484378686692653566, 8.580948081918716184188059735206, 10.63619638814011157114354690116, 11.24377525629143485462916570108

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