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Properties

Label 1-532-532.51-r0-0-0
Degree $1$
Conductor $532$
Sign $-0.0626 + 0.998i$
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  + (−0.939 + 0.342i)3-s + (0.766 + 0.642i)5-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.766 + 0.642i)17-s + (−0.173 − 0.984i)23-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (−0.173 − 0.984i)29-s + 31-s + (−0.766 − 0.642i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (0.766 + 0.642i)5-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.766 + 0.642i)17-s + (−0.173 − 0.984i)23-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (−0.173 − 0.984i)29-s + 31-s + (−0.766 − 0.642i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0626 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0626 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $-0.0626 + 0.998i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ -0.0626 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7465018372 + 0.7948566171i\)
\(L(\frac12)\) \(\approx\) \(0.7465018372 + 0.7948566171i\)
\(L(1)\) \(\approx\) \(0.8654774766 + 0.3333065189i\)
\(L(1)\) \(\approx\) \(0.8654774766 + 0.3333065189i\)

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 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−23.31347738025646432186393511234, −22.291555969705321227401234740450, −21.736412070132619790829007883673, −20.92612260140380223045495292110, −19.81121317559215379864825298972, −18.97867825709245828876574477667, −17.924249033277260109313401551876, −17.3990370302093209403231974125, −16.53997280944766135827557809285, −16.007033817015250213565871069131, −14.57248353791929856067551553065, −13.65619541548245511838026773168, −12.87604247417244934692116901966, −12.066776895328666574340322571897, −11.27067583790858269399553667159, −10.15172309409609310811013758738, −9.47674073083581141805665592334, −8.24913479911986116785180871396, −7.24371434831667367921698791636, −6.13457402002530534100298049632, −5.45527301987308568822164541617, −4.694389702022875996533581640894, −3.15716814722972175424368536554, −1.725970624880911353626917796944, −0.71123386380064209441331078181, 1.40559306926531119118363370292, 2.559820002841570497737628863746, 4.036123609695143243452790906987, 4.87475851941973084437598122025, 6.03080125882713879385386265662, 6.61619691693664627141850489957, 7.58085714843168028165724685790, 9.1752962415611868511883842316, 10.00991280181638345036300892582, 10.45499765589024337423813225912, 11.70557083050966644189602857800, 12.26034320446948050331855122373, 13.3659123406768914983654730997, 14.5551716550512181084109047349, 15.00365641096827569045318690315, 16.22639586830013764405500452901, 17.24492311983164069659588182420, 17.41170864365123130197962713398, 18.57710767346612680356914846907, 19.24225102876167507206598648701, 20.65856036206515021042022134571, 21.27066746569389749161928841077, 22.28247963069105540737167688074, 22.51743518385313839033155157346, 23.56552481600480579673469793527

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