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Properties

Label 1-547-547.233-r0-0-0
Degree $1$
Conductor $547$
Sign $0.0803 - 0.996i$
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  + (−0.632 − 0.774i)2-s + 3-s + (−0.200 + 0.979i)4-s + (−0.0402 − 0.999i)5-s + (−0.632 − 0.774i)6-s + (0.987 + 0.160i)7-s + (0.885 − 0.464i)8-s + 9-s + (−0.748 + 0.663i)10-s + (0.692 − 0.721i)11-s + (−0.200 + 0.979i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.0402 − 0.999i)15-s + (−0.919 − 0.391i)16-s + (0.278 − 0.960i)17-s + ⋯
L(s)  = 1  + (−0.632 − 0.774i)2-s + 3-s + (−0.200 + 0.979i)4-s + (−0.0402 − 0.999i)5-s + (−0.632 − 0.774i)6-s + (0.987 + 0.160i)7-s + (0.885 − 0.464i)8-s + 9-s + (−0.748 + 0.663i)10-s + (0.692 − 0.721i)11-s + (−0.200 + 0.979i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.0402 − 0.999i)15-s + (−0.919 − 0.391i)16-s + (0.278 − 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0803 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0803 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.0803 - 0.996i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.0803 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.189192410 - 1.097154964i\)
\(L(\frac12)\) \(\approx\) \(1.189192410 - 1.097154964i\)
\(L(1)\) \(\approx\) \(1.080707283 - 0.5739852695i\)
\(L(1)\) \(\approx\) \(1.080707283 - 0.5739852695i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.632 - 0.774i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.0402 - 0.999i)T \)
7 \( 1 + (0.987 + 0.160i)T \)
11 \( 1 + (0.692 - 0.721i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.278 - 0.960i)T \)
19 \( 1 + (-0.919 + 0.391i)T \)
23 \( 1 + (0.428 - 0.903i)T \)
29 \( 1 + (0.568 - 0.822i)T \)
31 \( 1 + (0.120 + 0.992i)T \)
37 \( 1 + (0.428 + 0.903i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.948 - 0.316i)T \)
47 \( 1 + (-0.996 - 0.0804i)T \)
53 \( 1 + (-0.845 + 0.534i)T \)
59 \( 1 + (-0.919 - 0.391i)T \)
61 \( 1 + (-0.632 - 0.774i)T \)
67 \( 1 + (0.948 - 0.316i)T \)
71 \( 1 + (0.799 + 0.600i)T \)
73 \( 1 + (-0.0402 + 0.999i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (-0.845 + 0.534i)T \)
89 \( 1 + (-0.354 - 0.935i)T \)
97 \( 1 + (0.692 + 0.721i)T \)
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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

−23.82061894189683438071883414705, −22.94549254257733731634384769042, −21.91439254847538395587950573397, −20.98396501447512411198765489098, −19.77036973314253166876571070110, −19.50387560813377287064741395411, −18.46458907479722817853609780232, −17.65028832722798781121519525843, −17.14370060776830677574933188584, −15.60762393400542602106278069819, −14.88415364282502955104662680165, −14.68805335330009606069381044651, −13.797946305594042302573583351860, −12.63061207263283199902659989126, −11.13181039368415177876426631503, −10.39354881405498779401876887353, −9.58125159993567003442426730933, −8.57581318722871105679821828682, −7.697071759181648326980083962847, −7.225064481191538084025056678631, −6.15692340042406776727195603770, −4.81397609856951118655446207034, −3.80799514841293876777816737936, −2.37337486261938314024725355125, −1.47823427846926767239179561604, 1.088175386138896051107943192, 1.90797587764109626442913550716, 2.9791449004183659908241824693, 4.28388056360012611877155383499, 4.75128223744530790878065081629, 6.68789831910409040927286358359, 7.93654632865076302020490220073, 8.46476682401657894816153745359, 9.1359160297737700053543311313, 9.90869775512374949461983516043, 11.17925603672787089932255979726, 11.995538137075247486008252049069, 12.72596636972415154780799592904, 13.85946066996750733221301620192, 14.35571614877738134232695962730, 15.70538284525577201803142025256, 16.67291962213363116358024395046, 17.24459666995141639095491498744, 18.50177715903318928979335328109, 19.0577092111619017454763950238, 19.93620555441386459170534099742, 20.56958177377787472921547476849, 21.367945050315284517862321389394, 21.65589383823403363351087133088, 23.232963393036965941507694568964

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