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Properties

Label 1-532-532.527-r0-0-0
Degree $1$
Conductor $532$
Sign $0.182 - 0.983i$
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  + (0.766 − 0.642i)3-s + (0.766 − 0.642i)5-s + (0.173 − 0.984i)9-s − 11-s + (0.939 − 0.342i)13-s + (0.173 − 0.984i)15-s + (0.173 + 0.984i)17-s + (0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.766 − 0.642i)29-s + (−0.5 − 0.866i)31-s + (−0.766 + 0.642i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (0.766 − 0.642i)5-s + (0.173 − 0.984i)9-s − 11-s + (0.939 − 0.342i)13-s + (0.173 − 0.984i)15-s + (0.173 + 0.984i)17-s + (0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.766 − 0.642i)29-s + (−0.5 − 0.866i)31-s + (−0.766 + 0.642i)33-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.182 - 0.983i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ 0.182 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.536766780 - 1.277436238i\)
\(L(\frac12)\) \(\approx\) \(1.536766780 - 1.277436238i\)
\(L(1)\) \(\approx\) \(1.389283306 - 0.5739959652i\)
\(L(1)\) \(\approx\) \(1.389283306 - 0.5739959652i\)

Euler product

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

−23.49775019871327892582922712877, −22.74298833539337542025894474887, −21.73676386342138535574903987976, −21.11387107051968145821375185435, −20.57999509503330337659712297664, −19.46840751679764956004122908139, −18.49983731854860691240145480162, −18.03549608058284562941884550539, −16.66996795951617084173371984291, −15.951391781525020641153517064675, −15.08566168374306107174106774893, −14.22906883984552984191253692487, −13.5780055703185376973475110268, −12.80654067419941462231559749338, −11.06249976625206542945167096972, −10.74040025735181472841790955187, −9.562184058874585339564831058704, −9.05555070982390383174565395045, −7.83507029523356400585852621107, −6.975088087380241309958072393789, −5.67544152974073526548366650776, −4.86389273500401182039470543421, −3.47762418307817248794207955209, −2.78580908167815806685302749731, −1.70223384915194020841172941322, 1.02821122485744299820779926445, 2.04881439852285057352410211554, 3.0474031159450821664013103841, 4.272231991989890213003353356137, 5.61295939940880880446400575779, 6.29779223156379600928400644997, 7.62669944185803348838868502639, 8.34583385637411961774535005735, 9.142800209294021081861986837978, 10.085966619479872455753675122650, 11.121584503692231365595228214961, 12.518254609684064102445120340635, 13.10915034959448589798289584166, 13.565142787976660215068848211557, 14.719891392112249866317605330188, 15.47355520476374269064457036268, 16.598092670291610874101002464842, 17.496618006673948670977048115768, 18.332661219496038128821965152926, 18.96570774396933521762088725087, 20.0541426616670796374253173577, 20.900630808107049435298601289912, 21.12254064085585892963975570883, 22.4810679108997110564148436066, 23.61388098243204963745657926966

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