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Properties

Label 1-532-532.303-r0-0-0
Degree $1$
Conductor $532$
Sign $-0.895 + 0.444i$
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  + (−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

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06601812957 + 0.2818140891i\)
\(L(\frac12)\) \(\approx\) \(0.06601812957 + 0.2818140891i\)
\(L(1)\) \(\approx\) \(0.6247622464 + 0.1157391676i\)
\(L(1)\) \(\approx\) \(0.6247622464 + 0.1157391676i\)

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

−22.92749312241931402857201037173, −22.541721929101795104651249143289, −21.80291075783481160357568573189, −20.21123832252171398134211597510, −19.76731992766924095999498729720, −18.6378079571581209297564447280, −18.29637495946304937614486221421, −17.23181884647965897399633728887, −16.578762732185347127246949537948, −15.237671070700496049573186588914, −14.63889267683166509422234978915, −13.65947310872802176098447220583, −12.62338404580540089823875389781, −11.87800825717086168375886388449, −11.19175815340865036694658628118, −10.221605094339495197653422134520, −9.10966253067953668845571781396, −7.73370789808867938771995427077, −7.15209667511041768346307437988, −6.51342251203297494308737096472, −5.247473960804486451098068722456, −4.21210568457401116080116418328, −2.75952467154092144822792860785, −1.93921381223955649410864974061, −0.165477521250542596541130516220, 1.38841303424923752295736507827, 3.22286924142962025370033226933, 4.07703197266772149901781481375, 4.99879852387997315658066060865, 5.769599144285538908346265218903, 6.97041600874877999706710382532, 8.27286398555495895685304155005, 9.03141103763543614997936236152, 9.83805371164574265657578413696, 10.984566540471021701740718291060, 11.65237655139595589310231866661, 12.48078594257674426880802162518, 13.48283884003268687942700974791, 14.77020137808521568767752161522, 15.349786777855022348324229510950, 16.36407121089021409743607370217, 16.907950871471791714954712833448, 17.54432656803858569110150422311, 18.949489205508623654729575213515, 19.807552216497834144163074712656, 20.42431788746501810071108792238, 21.61767058443347370133732592073, 21.85865383117435078741876379745, 22.987778812485892115156221745116, 23.85853517182838280416249869302

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