Properties

Label 2-536-536.123-c0-0-0
Degree $2$
Conductor $536$
Sign $0.539 + 0.842i$
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  + (−0.327 − 0.945i)2-s + (0.0395 + 0.0865i)3-s + (−0.786 + 0.618i)4-s + (0.0688 − 0.0656i)6-s + (0.841 + 0.540i)8-s + (0.648 − 0.748i)9-s + (1.21 + 1.16i)11-s + (−0.0845 − 0.0436i)12-s + (0.235 − 0.971i)16-s + (−1.45 − 1.14i)17-s + (−0.919 − 0.368i)18-s + (0.462 − 0.0892i)19-s + (0.698 − 1.53i)22-s + (−0.0135 + 0.0941i)24-s + (0.841 − 0.540i)25-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)2-s + (0.0395 + 0.0865i)3-s + (−0.786 + 0.618i)4-s + (0.0688 − 0.0656i)6-s + (0.841 + 0.540i)8-s + (0.648 − 0.748i)9-s + (1.21 + 1.16i)11-s + (−0.0845 − 0.0436i)12-s + (0.235 − 0.971i)16-s + (−1.45 − 1.14i)17-s + (−0.919 − 0.368i)18-s + (0.462 − 0.0892i)19-s + (0.698 − 1.53i)22-s + (−0.0135 + 0.0941i)24-s + (0.841 − 0.540i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $0.539 + 0.842i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ 0.539 + 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7783747327\)
\(L(\frac12)\) \(\approx\) \(0.7783747327\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.327 + 0.945i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
good3 \( 1 + (-0.0395 - 0.0865i)T + (-0.654 + 0.755i)T^{2} \)
5 \( 1 + (-0.841 + 0.540i)T^{2} \)
7 \( 1 + (-0.928 - 0.371i)T^{2} \)
11 \( 1 + (-1.21 - 1.16i)T + (0.0475 + 0.998i)T^{2} \)
13 \( 1 + (0.995 - 0.0950i)T^{2} \)
17 \( 1 + (1.45 + 1.14i)T + (0.235 + 0.971i)T^{2} \)
19 \( 1 + (-0.462 + 0.0892i)T + (0.928 - 0.371i)T^{2} \)
23 \( 1 + (-0.981 - 0.189i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.995 + 0.0950i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (1.21 - 0.486i)T + (0.723 - 0.690i)T^{2} \)
43 \( 1 + (0.205 - 1.43i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.327 + 0.945i)T^{2} \)
53 \( 1 + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (1.49 + 0.961i)T + (0.415 + 0.909i)T^{2} \)
61 \( 1 + (-0.0475 + 0.998i)T^{2} \)
71 \( 1 + (-0.235 + 0.971i)T^{2} \)
73 \( 1 + (1.13 - 1.08i)T + (0.0475 - 0.998i)T^{2} \)
79 \( 1 + (-0.580 - 0.814i)T^{2} \)
83 \( 1 + (0.452 - 1.86i)T + (-0.888 - 0.458i)T^{2} \)
89 \( 1 + (0.271 - 0.595i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.96009578207721548822514061368, −9.808305829126363559558629359406, −9.423594269421380526699330264439, −8.627693724883521559266860593875, −7.25459964089033471729857641194, −6.62281750360340739944053354935, −4.75168784868128084964356707600, −4.18829502873937277372554100223, −2.86608013324098290884031516426, −1.44849931297916636453214105704, 1.58151440606263223617628536540, 3.71584161582107853069435804230, 4.71678743691970897790020736100, 5.88571163108810616532870743301, 6.69294082642769237934160408530, 7.50860921096591077398211806769, 8.728646913997560056919080487925, 8.924974624162640347282830430688, 10.32432908238191490544795429345, 10.86829446931286942902845681604

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