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Properties

Label 1-572-572.499-r0-0-0
Degree $1$
Conductor $572$
Sign $0.00361 + 0.999i$
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  + (0.809 + 0.587i)3-s + (−0.951 − 0.309i)5-s + (0.587 + 0.809i)7-s + (0.309 + 0.951i)9-s + (−0.587 − 0.809i)15-s + (−0.309 + 0.951i)17-s + (0.587 − 0.809i)19-s + i·21-s + 23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.951 + 0.309i)31-s + (−0.309 − 0.951i)35-s + (−0.587 − 0.809i)37-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (−0.951 − 0.309i)5-s + (0.587 + 0.809i)7-s + (0.309 + 0.951i)9-s + (−0.587 − 0.809i)15-s + (−0.309 + 0.951i)17-s + (0.587 − 0.809i)19-s + i·21-s + 23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.951 + 0.309i)31-s + (−0.309 − 0.951i)35-s + (−0.587 − 0.809i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00361 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00361 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.00361 + 0.999i$
Analytic conductor: \(2.65635\)
Root analytic conductor: \(2.65635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 572,\ (0:\ ),\ 0.00361 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.084414390 + 1.088342693i\)
\(L(\frac12)\) \(\approx\) \(1.084414390 + 1.088342693i\)
\(L(1)\) \(\approx\) \(1.149532856 + 0.4444219076i\)
\(L(1)\) \(\approx\) \(1.149532856 + 0.4444219076i\)

Euler product

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

−23.08342440003856481108535239510, −22.502490004181288991530932512042, −20.95777300362269239941915171007, −20.472005441413188000369978065040, −19.76981259949123002487766445035, −18.84602667577164065727973201319, −18.31622787490756045401469817925, −17.25438852846962354722725732707, −16.22712960957179865536202425069, −15.23794916290910509593887315830, −14.54416601577537962581126682017, −13.795338998022639480000382563, −12.94083360555971729639384363109, −11.85464593907112300252629913247, −11.21519465435302137767087336376, −10.07871850541298149979594866046, −8.97495766494854102733217479197, −8.036000320192703017911338114666, −7.36867788643820143315193124570, −6.78685510999940049516434839698, −5.16155071486867566656735642443, −3.96027964994721926830659460189, −3.31146200302147338385870310754, −2.03813370345753780635285503331, −0.7569561609673351196011023347, 1.557924113937658685811352498721, 2.788102608351559642914979889924, 3.72725184330578612286741553062, 4.69578447697317363202632721629, 5.46077168171693512210262190854, 7.1164220481721294585841200834, 7.940608085835137998090280137158, 8.82514643682445990183499905882, 9.244540213905495131257256943063, 10.75541303944858652388381018726, 11.28766626328437382450079903722, 12.45296791480134995948683190720, 13.20892683754104190015021502243, 14.501824957289104373024930775239, 15.02197636806308062181180412930, 15.71587590059456365599373656548, 16.474141793455633872039125812070, 17.60628128493026679709946775727, 18.70848111024781931358579552021, 19.39885496750071120888578093962, 20.14049394869776624697923121774, 20.897937089368375690900385721, 21.72384521644873983994141588658, 22.40709318967465806080847397712, 23.61993756732887832003356652470

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