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Properties

Label 1-532-532.439-r1-0-0
Degree $1$
Conductor $532$
Sign $0.149 + 0.988i$
Analytic cond. $57.1713$
Root an. cond. $57.1713$
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  + (0.939 + 0.342i)3-s + (0.939 + 0.342i)5-s + (0.766 + 0.642i)9-s − 11-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.766 + 0.642i)17-s + (−0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (0.5 + 0.866i)27-s + (0.939 − 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.939 − 0.342i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)3-s + (0.939 + 0.342i)5-s + (0.766 + 0.642i)9-s − 11-s + (0.173 − 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.766 + 0.642i)17-s + (−0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (0.5 + 0.866i)27-s + (0.939 − 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.939 − 0.342i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.149 + 0.988i$
Analytic conductor: \(57.1713\)
Root analytic conductor: \(57.1713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (1:\ ),\ 0.149 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.462343267 + 2.118336062i\)
\(L(\frac12)\) \(\approx\) \(2.462343267 + 2.118336062i\)
\(L(1)\) \(\approx\) \(1.604375142 + 0.4942496000i\)
\(L(1)\) \(\approx\) \(1.604375142 + 0.4942496000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 - T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.25386553680961449326762478672, −21.959668298364840660502012219618, −21.2345246043513932783244795872, −20.58430341662720766886604454290, −19.88139846205055004880136294819, −18.62299466397921709227339094173, −18.31559360416397591213524310392, −17.23826240623913463605891641754, −16.18471477629587685632133824390, −15.395851777723969391009094870743, −14.20585888180226719926503631198, −13.7374774447295825443431150323, −12.9682101179734022643575012832, −12.11422735261813338221336043631, −10.73630985894809527140141439099, −9.81301994521679240730951354843, −8.99347018368418050334261585920, −8.3115362458256745505524456610, −7.11909298522714684679819542128, −6.322558994926394706945341124325, −5.05931282787414894195418902059, −4.07077774623653242251774478829, −2.56109161256753444284599372029, −2.1139205700762998723080154492, −0.69208390621714710604627039068, 1.37835480453087652513634275414, 2.55561456137961349820764950604, 3.1663902315008590093506031054, 4.55392595540526874340257551341, 5.53332207991120115737840072892, 6.60937067468052367065380189286, 7.817615877382735460858274378447, 8.49582014399692270786818612196, 9.62560483602543049981010469369, 10.270248261083003447662333181264, 10.97165749162219714496065340341, 12.60242652939148073361264781621, 13.42206744908577553350879868568, 13.853244542468087720192549651207, 15.1326264616208996856625572833, 15.43271412573357280109345692706, 16.61145835626604060072349851785, 17.81270559157240469839834437078, 18.23167917996283967791505643729, 19.43011563086496384464720473430, 20.09068147765190589834137878352, 21.13282583289523282054587939037, 21.48853380905091086611660602039, 22.418666020214963641399850936062, 23.459421083310067412500974714831

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