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Properties

Label 1-532-532.487-r0-0-0
Degree $1$
Conductor $532$
Sign $0.987 - 0.160i$
Analytic cond. $2.47059$
Root an. cond. $2.47059$
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  + 3-s + (−0.5 − 0.866i)5-s + 9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + 17-s − 23-s + (−0.5 + 0.866i)25-s + 27-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + 3-s + (−0.5 − 0.866i)5-s + 9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + 17-s − 23-s + (−0.5 + 0.866i)25-s + 27-s + (0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.987 - 0.160i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ 0.987 - 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.948048120 - 0.1574197561i\)
\(L(\frac12)\) \(\approx\) \(1.948048120 - 0.1574197561i\)
\(L(1)\) \(\approx\) \(1.466137645 - 0.09034432641i\)
\(L(1)\) \(\approx\) \(1.466137645 - 0.09034432641i\)

Euler product

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

−23.50128847289898793296455324922, −22.63404730223888376110182215251, −21.7192473546535093715949870093, −21.000373633103577701325115244682, −19.89109793303561359534017588014, −19.4291244542098024089141572408, −18.54128992783533637365990534409, −17.921556327835457647989957231991, −16.439435517831624119297374833962, −15.73453367063937668403247682542, −14.82437575687032950929186040180, −14.216088261675034459259890525771, −13.43469661530141219474942044194, −12.33059803081722534295992408910, −11.33504229808283351063972016497, −10.3669761571492862881520355954, −9.5948426643543324824630894551, −8.2392409863903425530659829774, −7.96292327366264205970965644993, −6.758149269556948871107054850058, −5.823026057272598166937901376687, −4.21871448109231741631181926057, −3.365835924937705914796765892094, −2.73675977566910449178373046080, −1.20522955129651093806235454468, 1.26229914763206423659638602991, 2.21073896123360745322430659166, 3.78202717357399741367067743183, 4.17469536738232710931048750464, 5.44129712034801129948743103663, 6.89079643304739953139782388975, 7.70955184952422914517406868172, 8.60236387168330480653557144279, 9.32287136419101639345051966119, 10.121445861610799172585121523532, 11.56941884848341242763229201273, 12.375456590807439288428509529784, 13.10768309520584544462158307338, 14.20491361064623374292175899826, 14.75682946437380986918523330418, 15.93410904315478388103144565278, 16.36368544405503148125499399035, 17.55581379354556678290733051801, 18.61916503373902323829231686010, 19.417102912883264454929886700218, 20.11411521719997950162608918507, 20.782376193619384453097579033087, 21.486783983101865024180572925715, 22.655971109591383993831844160591, 23.74761308556014776092345125949

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