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Properties

Label 1-532-532.75-r1-0-0
Degree $1$
Conductor $532$
Sign $0.386 - 0.922i$
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  + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + 13-s + 15-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s − 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  + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + 13-s + 15-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(57.1713\)
Root analytic conductor: \(57.1713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (1:\ ),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.458910717 - 1.635622878i\)
\(L(\frac12)\) \(\approx\) \(2.458910717 - 1.635622878i\)
\(L(1)\) \(\approx\) \(1.456417057 - 0.4234187814i\)
\(L(1)\) \(\approx\) \(1.456417057 - 0.4234187814i\)

Euler product

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

−23.2745062885729899311444928397, −22.53244847204950014641758844933, −21.41563960087584791565221617124, −21.000019320703389838156078494878, −20.18957545887127675564652342957, −19.530325538679051963098869419711, −18.31674870974672017386160250404, −17.24771081718180145552681330631, −16.6230813307837980292929600927, −15.81738342057345321029175792542, −14.86791790645419466456380381352, −14.14830861028754334160287046170, −13.12500277321695683283270559397, −12.399170726262167736568993734707, −11.11159572188693106792982026894, −10.23132238575035637711999092988, −9.36928691719433110574176294320, −8.7235507817417990158307075965, −7.85272020934616136189263001443, −6.37738935384640645736748971540, −5.37946401988058768323089134226, −4.43534578648172910766070791133, −3.660227304670908317264592835470, −2.265715603304491725534829249497, −1.18108173106411191107240322737, 0.76817880360075269072991668117, 1.8508074321749669940417186736, 3.023619691191698678642960581795, 3.65797067259718093737097895225, 5.55561664625982683309744084313, 6.304125361666088523137467870, 7.12498268316748029593760905515, 8.06618446055404636002091096570, 9.05843606978040039148207875567, 9.89409017967878718758526918308, 11.28922982748519087360696140114, 11.63589116072753503620969680125, 13.23273584467944589245287293599, 13.533102675098253812273255569128, 14.44002369478388167416045265586, 15.15996598887759214284354191107, 16.40470545909733500916388962970, 17.38474929140171778651492790179, 18.31312072993021992577462865861, 18.76460839037912375878746234266, 19.52391651028829249647004935782, 20.63242529793905668711900231304, 21.31299346227226133081535465426, 22.398803695569247117104351986094, 23.11305169860233728597138489999

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