Properties

Label 1-547-547.44-r0-0-0
Degree $1$
Conductor $547$
Sign $0.742 + 0.670i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (0.675 + 0.736i)2-s + (−0.900 − 0.433i)3-s + (−0.0862 + 0.996i)4-s + (−0.792 − 0.609i)5-s + (−0.289 − 0.957i)6-s + (−0.868 − 0.495i)7-s + (−0.792 + 0.609i)8-s + (0.623 + 0.781i)9-s + (−0.0862 − 0.996i)10-s + (0.120 + 0.992i)11-s + (0.509 − 0.860i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.449 + 0.893i)15-s + (−0.985 − 0.171i)16-s + (0.851 − 0.524i)17-s + ⋯
L(s)  = 1  + (0.675 + 0.736i)2-s + (−0.900 − 0.433i)3-s + (−0.0862 + 0.996i)4-s + (−0.792 − 0.609i)5-s + (−0.289 − 0.957i)6-s + (−0.868 − 0.495i)7-s + (−0.792 + 0.609i)8-s + (0.623 + 0.781i)9-s + (−0.0862 − 0.996i)10-s + (0.120 + 0.992i)11-s + (0.509 − 0.860i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (0.449 + 0.893i)15-s + (−0.985 − 0.171i)16-s + (0.851 − 0.524i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.742 + 0.670i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.742 + 0.670i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9066478699 + 0.3486945201i\)
\(L(\frac12)\) \(\approx\) \(0.9066478699 + 0.3486945201i\)
\(L(1)\) \(\approx\) \(0.8600321671 + 0.2238327763i\)
\(L(1)\) \(\approx\) \(0.8600321671 + 0.2238327763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.675 + 0.736i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.792 - 0.609i)T \)
7 \( 1 + (-0.868 - 0.495i)T \)
11 \( 1 + (0.120 + 0.992i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.851 - 0.524i)T \)
19 \( 1 + (-0.479 + 0.877i)T \)
23 \( 1 + (-0.650 - 0.759i)T \)
29 \( 1 + (0.915 - 0.402i)T \)
31 \( 1 + (0.962 - 0.272i)T \)
37 \( 1 + (0.188 + 0.982i)T \)
41 \( 1 + T \)
43 \( 1 + (0.725 + 0.688i)T \)
47 \( 1 + (0.885 + 0.464i)T \)
53 \( 1 + (0.449 + 0.893i)T \)
59 \( 1 + (-0.748 + 0.663i)T \)
61 \( 1 + (0.997 - 0.0689i)T \)
67 \( 1 + (0.509 - 0.860i)T \)
71 \( 1 + (0.0517 - 0.998i)T \)
73 \( 1 + (0.449 - 0.893i)T \)
79 \( 1 + (0.725 + 0.688i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 + (0.997 + 0.0689i)T \)
97 \( 1 + (-0.994 + 0.103i)T \)
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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

−23.29041366426856758552500223077, −22.27419409066581604904973643334, −21.70709054690038677366079683620, −21.277212587897023855717427886837, −19.75014362799937424585923695709, −19.167890291091092648470631281652, −18.632652339849467617940959212897, −17.43977433151768294444854205832, −16.0651646531804686180561885626, −15.84282498196349344748440465009, −14.76117410737687839233500539052, −13.90423926973750463082534103386, −12.66840284907284331479767467520, −11.98936762027571257282124045673, −11.36042132131957254500889484744, −10.56787664450453529379284346421, −9.732907083636820421568659670081, −8.73055887443827824090145183488, −6.9715241268739296731433902581, −6.2534235115247553033874280229, −5.44805242446599469877270034295, −4.17436280934996812814223990119, −3.58033350550087757214171380273, −2.516501123684416555361936578208, −0.72134495418857480902089792685, 0.79191719522333011812497967807, 2.74336887320781150714108670980, 4.091080684994348202607784868546, 4.69373783272905547167130001251, 5.8005757802555793549958787013, 6.59274873630527115410593947951, 7.62249559778866089326419711370, 8.02110103103190855333861805413, 9.63367803111965490752423631072, 10.606997408259259233674618324193, 12.13197001644877428441889186504, 12.25025102672567533192139869328, 13.01746320087047160316877282807, 14.02675958994001782448475509504, 15.24808863364901264532408235697, 15.93635373058781374877080865541, 16.68104182571249791951141683082, 17.23036356568079215897977045594, 18.2134417649107855503049768126, 19.28197836414664325467786033953, 20.253053164349356545233227240823, 21.03818401437029629871689178517, 22.41147560797151456332509990303, 22.905583283786968344673365298364, 23.21341793459892716095783015119

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