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Properties

Label 1-547-547.3-r1-0-0
Degree $1$
Conductor $547$
Sign $-0.933 - 0.359i$
Analytic cond. $58.7833$
Root an. cond. $58.7833$
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  + (0.222 − 0.974i)2-s + (0.900 + 0.433i)3-s + (−0.900 − 0.433i)4-s + (−0.623 − 0.781i)5-s + (0.623 − 0.781i)6-s + (0.900 − 0.433i)7-s + (−0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + 11-s + (−0.623 − 0.781i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + (−0.623 − 0.781i)17-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (0.900 + 0.433i)3-s + (−0.900 − 0.433i)4-s + (−0.623 − 0.781i)5-s + (0.623 − 0.781i)6-s + (0.900 − 0.433i)7-s + (−0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + 11-s + (−0.623 − 0.781i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + (−0.623 − 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.933 - 0.359i$
Analytic conductor: \(58.7833\)
Root analytic conductor: \(58.7833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (1:\ ),\ -0.933 - 0.359i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4843349353 - 2.602047015i\)
\(L(\frac12)\) \(\approx\) \(0.4843349353 - 2.602047015i\)
\(L(1)\) \(\approx\) \(1.128493829 - 0.9573745970i\)
\(L(1)\) \(\approx\) \(1.128493829 - 0.9573745970i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.222 - 0.974i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.623 - 0.781i)T \)
19 \( 1 + (-0.222 - 0.974i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (0.900 + 0.433i)T \)
37 \( 1 + (-0.623 - 0.781i)T \)
41 \( 1 - T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (0.900 - 0.433i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−23.89949855775756398380697274636, −22.86132832585651164836071297136, −21.94807283776277396484213515714, −21.24967779046445636310964154182, −20.09320176526869472911183245789, −18.94487927076932842879088690641, −18.712286265372283756726957519898, −17.65352304761068235683869888166, −16.79199802666994937180751159572, −15.54511903180755780291937971785, −14.9263389999477254447953972821, −14.323900565898390311171363120209, −13.83056431044224227279966130836, −12.36137313444720260048259336830, −11.90845287924720872694202695295, −10.48083475161922442475652877526, −9.129647736317389542298238072668, −8.48870547218860577601080949180, −7.77025808240390682076698501632, −6.76279141912232290257327230439, −6.26410762140544699852000053585, −4.49832634218493080719613488342, −3.96249705178385368453918653140, −2.72824677909397824247996377541, −1.428472618703770391085867485183, 0.568224409160992442932258516662, 1.59018272531317089239866450455, 2.803731846212163576222928090134, 3.86539468314592656704139921108, 4.56171952181668403593698962280, 5.24736790576168659806043387800, 7.231604929876859074505933756305, 8.26165226900937115773347132168, 8.899432472620633891078372586551, 9.70692721741902338582176275140, 10.80767403829055767223329914634, 11.55172579509541011910093611565, 12.453646872849102749231388177302, 13.53996800518305046204672755223, 13.99248068565006723367372052830, 15.139917548958180537949094834, 15.64905343423028477359178484311, 17.166317306427077537480330526948, 17.71577333908223036127784067395, 19.14279376847733976706821172709, 19.73648342391122822658960468500, 20.27058313828618508753199737934, 20.90827085344244062146528793291, 21.72264454837715476920641421766, 22.62280519982780012815517075355

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