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Properties

Label 1-57-57.17-r1-0-0
Degree $1$
Conductor $57$
Sign $-0.158 + 0.987i$
Analytic cond. $6.12550$
Root an. cond. $6.12550$
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)2-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)10-s + (0.5 + 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 20-s + (0.173 + 0.984i)22-s + (−0.766 − 0.642i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)10-s + (0.5 + 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s − 20-s + (0.173 + 0.984i)22-s + (−0.766 − 0.642i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $-0.158 + 0.987i$
Analytic conductor: \(6.12550\)
Root analytic conductor: \(6.12550\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 57,\ (1:\ ),\ -0.158 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.487603150 + 1.744667923i\)
\(L(\frac12)\) \(\approx\) \(1.487603150 + 1.744667923i\)
\(L(1)\) \(\approx\) \(1.432338725 + 0.7970462988i\)
\(L(1)\) \(\approx\) \(1.432338725 + 0.7970462988i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−32.18046202254754291486752433763, −31.41615456825326222497843099063, −30.126998069486563292680013303659, −29.24371174985357504416179301991, −28.06689896494682738041684780111, −26.846100467207790934130025006603, −25.28971718656723079210779019360, −23.89325505950998263047796475042, −23.45312389681413586728279714737, −22.09628043401327683107690731479, −20.87431964141885983516419997389, −19.80855512113806804760314718792, −19.02311024288493592671242525301, −16.627034487874237354771434520054, −16.05603381664387255845376518415, −14.384008030660038420994831631998, −13.39348842337175135584333154712, −12.09285028396950352242371669084, −11.1312029374982435726857893126, −9.51983830615276684184325291224, −7.61334218502991867023541367650, −6.14580844086326038786558491921, −4.42903168576556116253493824770, −3.43586172547184364963336872634, −1.02652290762661720039895217833, 2.6708859450365844379171057466, 3.96161010901750841529312158243, 5.685943132377520305179438358776, 6.9647668272336937646989893728, 8.262335110489794748759976652797, 10.34996817506058062523931927079, 11.9363996521862985190752233335, 12.608991274649940151534949771049, 14.38448227990265936168596498852, 15.24854935876453587352929594401, 16.15854235941930140733366736820, 17.82003742197605010744012566835, 19.29797257377328065230237966471, 20.43999517968585527682271201544, 21.973427745839588717970703241394, 22.68472787282159763879324566920, 23.62775831544872019236406510771, 25.126560719700477676177448943145, 25.76709043411188836528727696592, 27.29862724930433629416499058668, 28.57156754324906678876318055774, 30.137236741982695865441997673305, 30.71055795379032466269711384273, 31.90347454065590589624427307831, 32.72351741331756084692814852070

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