Properties

Label 1-547-547.97-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.792 - 0.609i$
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  + (−0.131 − 0.991i)2-s + (−0.222 − 0.974i)3-s + (−0.965 + 0.261i)4-s + (−0.991 + 0.126i)5-s + (−0.937 + 0.349i)6-s + (0.874 + 0.484i)7-s + (0.386 + 0.922i)8-s + (−0.900 + 0.433i)9-s + (0.256 + 0.966i)10-s + (0.987 + 0.160i)11-s + (0.469 + 0.882i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (0.343 + 0.939i)15-s + (0.863 − 0.504i)16-s + (0.905 + 0.423i)17-s + ⋯
L(s)  = 1  + (−0.131 − 0.991i)2-s + (−0.222 − 0.974i)3-s + (−0.965 + 0.261i)4-s + (−0.991 + 0.126i)5-s + (−0.937 + 0.349i)6-s + (0.874 + 0.484i)7-s + (0.386 + 0.922i)8-s + (−0.900 + 0.433i)9-s + (0.256 + 0.966i)10-s + (0.987 + 0.160i)11-s + (0.469 + 0.882i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (0.343 + 0.939i)15-s + (0.863 − 0.504i)16-s + (0.905 + 0.423i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.792 - 0.609i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.792 - 0.609i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2725352758 - 0.8019774482i\)
\(L(\frac12)\) \(\approx\) \(0.2725352758 - 0.8019774482i\)
\(L(1)\) \(\approx\) \(0.5711000976 - 0.5226517042i\)
\(L(1)\) \(\approx\) \(0.5711000976 - 0.5226517042i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.131 - 0.991i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.991 + 0.126i)T \)
7 \( 1 + (0.874 + 0.484i)T \)
11 \( 1 + (0.987 + 0.160i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (0.905 + 0.423i)T \)
19 \( 1 + (-0.558 - 0.829i)T \)
23 \( 1 + (-0.880 - 0.474i)T \)
29 \( 1 + (0.322 - 0.946i)T \)
31 \( 1 + (0.675 - 0.736i)T \)
37 \( 1 + (0.998 - 0.0460i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.332 - 0.942i)T \)
47 \( 1 + (0.799 - 0.600i)T \)
53 \( 1 + (0.641 - 0.767i)T \)
59 \( 1 + (-0.996 + 0.0804i)T \)
61 \( 1 + (-0.311 + 0.950i)T \)
67 \( 1 + (0.529 - 0.848i)T \)
71 \( 1 + (-0.778 - 0.627i)T \)
73 \( 1 + (0.343 - 0.939i)T \)
79 \( 1 + (-0.650 + 0.759i)T \)
83 \( 1 + (-0.200 - 0.979i)T \)
89 \( 1 + (0.978 + 0.205i)T \)
97 \( 1 + (0.740 + 0.671i)T \)
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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

−23.52502703614551414466358018713, −23.15603114540379706054159822367, −22.16692838041637407528546539725, −21.45638409149687411820616855942, −20.24469928950394031915534258683, −19.62562898306198255843074868021, −18.52577220432725005908848291077, −17.37774250046930557023708620452, −16.865443418870168618676829151571, −16.16770506557887548388151836026, −15.30003855354308667024366053034, −14.35148430774655223309689437781, −14.27746055370035711201900007431, −12.3832437637899755190836214047, −11.70427855219188875450669512206, −10.56952418800393935764298450332, −9.76598454623744659789024089306, −8.71461148608913688277432305857, −7.96626032577092094682495387347, −7.1323266289099367067716460720, −5.88654621979204405580210616214, −4.852362648740801619885646038325, −4.26277720859013379752160364438, −3.39746360746255098628221666694, −1.04798355521616576558010347520, 0.62926534812854443275440937793, 1.8784619143717365015386755923, 2.718926216437476528766030314, 4.09528203063326211766065557384, 4.90285804062204803866924886781, 6.234156088676073560293539070601, 7.532087074751114838217588261053, 8.132435815898272253518299620448, 8.97653326897058764391321075762, 10.27140834399151394064565137701, 11.37125698212484042802005677761, 11.96973458464021086582072068098, 12.23276729297130929871552366562, 13.46123003463452605924469686649, 14.49061763323057463030669422145, 15.011752429396586598431870498526, 16.801594151222882674527414148480, 17.29669149281054775582894051880, 18.30540283266466374205804579835, 18.94717923510948707426232747513, 19.66235997277652070494710646048, 20.18333474224828418658774514794, 21.44638139895580100559170412354, 22.203138868495629553441241473865, 22.98641783407496057012571256168

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