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Properties

Label 1-572-572.31-r0-0-0
Degree $1$
Conductor $572$
Sign $0.284 + 0.958i$
Analytic cond. $2.65635$
Root an. cond. $2.65635$
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  + (−0.309 + 0.951i)3-s + (0.587 + 0.809i)5-s + (0.951 − 0.309i)7-s + (−0.809 − 0.587i)9-s + (−0.951 + 0.309i)15-s + (0.809 − 0.587i)17-s + (0.951 + 0.309i)19-s + i·21-s + 23-s + (−0.309 + 0.951i)25-s + (0.809 − 0.587i)27-s + (0.309 + 0.951i)29-s + (0.587 − 0.809i)31-s + (0.809 + 0.587i)35-s + (−0.951 + 0.309i)37-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (0.587 + 0.809i)5-s + (0.951 − 0.309i)7-s + (−0.809 − 0.587i)9-s + (−0.951 + 0.309i)15-s + (0.809 − 0.587i)17-s + (0.951 + 0.309i)19-s + i·21-s + 23-s + (−0.309 + 0.951i)25-s + (0.809 − 0.587i)27-s + (0.309 + 0.951i)29-s + (0.587 − 0.809i)31-s + (0.809 + 0.587i)35-s + (−0.951 + 0.309i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.284 + 0.958i$
Analytic conductor: \(2.65635\)
Root analytic conductor: \(2.65635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 572,\ (0:\ ),\ 0.284 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.277415447 + 0.9533161735i\)
\(L(\frac12)\) \(\approx\) \(1.277415447 + 0.9533161735i\)
\(L(1)\) \(\approx\) \(1.121556329 + 0.4678406253i\)
\(L(1)\) \(\approx\) \(1.121556329 + 0.4678406253i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.951 + 0.309i)T \)
23 \( 1 + T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.587 - 0.809i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.951 - 0.309i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.587 - 0.809i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−23.302724069971558604113770911194, −22.282414189753809568487486493298, −21.25357203140309109829434488637, −20.72598158950037033102833743132, −19.64903484724904067350451837833, −18.84461811084368093281245824398, −17.86881516983125204419666162059, −17.39707746330648032096147295670, −16.65503540099782470306395005222, −15.52666352048589259239188806262, −14.30837472352564248734438917790, −13.738664994960825619892909345440, −12.76463537687466114392966066576, −12.08390927088836855501404484095, −11.30767522037389129663138981583, −10.19428486138075173861172167611, −8.98468232840986846854283953535, −8.24428259598809413660318124956, −7.42113447482258253639247996813, −6.22369677916355579753595498308, −5.376894236406241873615121532826, −4.72135178492318755179058866825, −2.960831134145192997234584806003, −1.74662526097825873243090198309, −1.065585530181516775047892490937, 1.30015887383398375351915712412, 2.79455039461493766698731217908, 3.6377449378207620839588857348, 4.94945658332473924264313969115, 5.479559184812463051410664099527, 6.692485843810762867770587338660, 7.66379399587967895180293339437, 8.85640284468791291530401030109, 9.83573698637404437665997711701, 10.46737536009314128308293815454, 11.29840240574452234492311792549, 11.97737622776879065724075081652, 13.51951477661254762237496969542, 14.342403793674538379263781190307, 14.8183910137124391450070247749, 15.83166582957455098675503504500, 16.785206103638909984950549592407, 17.52695019612795923572922125388, 18.20116702232160625204597496641, 19.164115976864665681670769611295, 20.60169971320524737180041941067, 20.833741286070175556139772193084, 21.760952599869749452465224441795, 22.53681746061733387566037924809, 23.13988347204455690276253766747

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