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Properties

Label 1-572-572.15-r0-0-0
Degree $1$
Conductor $572$
Sign $-0.744 - 0.667i$
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  + (−0.913 + 0.406i)3-s + (−0.951 − 0.309i)5-s + (0.406 − 0.913i)7-s + (0.669 − 0.743i)9-s + (0.994 − 0.104i)15-s + (0.978 − 0.207i)17-s + (−0.994 − 0.104i)19-s + i·21-s + (−0.5 + 0.866i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.104 − 0.994i)29-s + (−0.951 + 0.309i)31-s + (−0.669 + 0.743i)35-s + (0.994 − 0.104i)37-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)3-s + (−0.951 − 0.309i)5-s + (0.406 − 0.913i)7-s + (0.669 − 0.743i)9-s + (0.994 − 0.104i)15-s + (0.978 − 0.207i)17-s + (−0.994 − 0.104i)19-s + i·21-s + (−0.5 + 0.866i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.104 − 0.994i)29-s + (−0.951 + 0.309i)31-s + (−0.669 + 0.743i)35-s + (0.994 − 0.104i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(2.65635\)
Root analytic conductor: \(2.65635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 572,\ (0:\ ),\ -0.744 - 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1393024279 - 0.3639915492i\)
\(L(\frac12)\) \(\approx\) \(0.1393024279 - 0.3639915492i\)
\(L(1)\) \(\approx\) \(0.5987985108 - 0.1029160336i\)
\(L(1)\) \(\approx\) \(0.5987985108 - 0.1029160336i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (0.406 - 0.913i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (-0.994 - 0.104i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (-0.951 + 0.309i)T \)
37 \( 1 + (0.994 - 0.104i)T \)
41 \( 1 + (-0.406 - 0.913i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.406 - 0.913i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.207 - 0.978i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.743 - 0.669i)T \)
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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

−23.54439347935567895044277327751, −22.86146978254658622630037950029, −21.96122465680705246926871549131, −21.36279424805844647450947792807, −20.10263481964034253180712559627, −19.15611143677740672163044643183, −18.46885986297142191276726111174, −17.993354392107115513297796294055, −16.63466429921125480754871982143, −16.27237509970571536191308911849, −15.00466079756733696375133479782, −14.59849854066048223235593791457, −13.00709344472772846831889494946, −12.3553598261374817514877948967, −11.63258226964998354434590248596, −10.950352664377213799575560663541, −10.00619367938150644442401126765, −8.53509312025152455030550165135, −7.889144559301470420749548612741, −6.85170879155517937623324553530, −5.970275329722433680182724082713, −5.01681916219032461974737671891, −4.04389290643098528127073552693, −2.684712987363331956421939047318, −1.43885345066523300770401715723, 0.246645182582179969368613483482, 1.48853535205266958839200312953, 3.49161178824195518256970794085, 4.196869506888449536930827067371, 5.00814677265183822550411952317, 6.08007120574567933437812218353, 7.269043153890641800825079673875, 7.876809348573477983147169693169, 9.15021075657300575488251697647, 10.22650207415073623646394831760, 10.957876061036152635180346602148, 11.72705383650475623689043832046, 12.45272633633217293506724351632, 13.49035004932403723303347359017, 14.69328449776875005715965208801, 15.435875734472745008847575787820, 16.39521128099001083432272750487, 16.88794256254143750890679299924, 17.71686232408991214220086309533, 18.748031249781365929338117815504, 19.64413803482051947185928643595, 20.55887640925553309807245809143, 21.207743476648241532436478070058, 22.20119135554025316486453846860, 23.19662033381724816951964060194

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