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Properties

Label 1-547-547.325-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.930 + 0.366i$
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  + (0.428 + 0.903i)2-s + 3-s + (−0.632 + 0.774i)4-s + (0.692 + 0.721i)5-s + (0.428 + 0.903i)6-s + (−0.996 + 0.0804i)7-s + (−0.970 − 0.239i)8-s + 9-s + (−0.354 + 0.935i)10-s + (−0.919 − 0.391i)11-s + (−0.632 + 0.774i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.692 + 0.721i)15-s + (−0.200 − 0.979i)16-s + (0.799 + 0.600i)17-s + ⋯
L(s)  = 1  + (0.428 + 0.903i)2-s + 3-s + (−0.632 + 0.774i)4-s + (0.692 + 0.721i)5-s + (0.428 + 0.903i)6-s + (−0.996 + 0.0804i)7-s + (−0.970 − 0.239i)8-s + 9-s + (−0.354 + 0.935i)10-s + (−0.919 − 0.391i)11-s + (−0.632 + 0.774i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.692 + 0.721i)15-s + (−0.200 − 0.979i)16-s + (0.799 + 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.930 + 0.366i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.930 + 0.366i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3577736405 + 1.887090018i\)
\(L(\frac12)\) \(\approx\) \(0.3577736405 + 1.887090018i\)
\(L(1)\) \(\approx\) \(1.072848121 + 1.092639477i\)
\(L(1)\) \(\approx\) \(1.072848121 + 1.092639477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.428 + 0.903i)T \)
3 \( 1 + T \)
5 \( 1 + (0.692 + 0.721i)T \)
7 \( 1 + (-0.996 + 0.0804i)T \)
11 \( 1 + (-0.919 - 0.391i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.799 + 0.600i)T \)
19 \( 1 + (-0.200 + 0.979i)T \)
23 \( 1 + (-0.845 - 0.534i)T \)
29 \( 1 + (0.885 + 0.464i)T \)
31 \( 1 + (-0.748 + 0.663i)T \)
37 \( 1 + (-0.845 + 0.534i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.987 + 0.160i)T \)
47 \( 1 + (-0.0402 - 0.999i)T \)
53 \( 1 + (0.278 - 0.960i)T \)
59 \( 1 + (-0.200 - 0.979i)T \)
61 \( 1 + (0.428 + 0.903i)T \)
67 \( 1 + (0.987 + 0.160i)T \)
71 \( 1 + (0.948 - 0.316i)T \)
73 \( 1 + (0.692 - 0.721i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (0.278 - 0.960i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (-0.919 + 0.391i)T \)
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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

−22.849536309460320791650760760349, −21.92174034061513511013294679600, −21.214528178510215186991070237749, −20.4125951010427940405232736031, −19.94982795761740598804161218410, −19.12038099300817804619580983419, −18.21881670220475995858340694039, −17.347899856571149064084211791543, −15.87915795847590293927100284846, −15.36835597043756081336461081690, −14.04001363965506341443460082264, −13.593238932999412686380157388905, −12.63046156937940481455879983333, −12.42814776120793269346992899163, −10.643527706065298258808592585769, −9.79924952841988584928518326604, −9.49182983376835987227881846286, −8.38739685520972367838187700319, −7.25092832082957661624159571565, −5.79813350311023531656353570936, −4.98456979593385213751789126441, −3.877207202304839294757455589212, −2.77513964966240725427415951794, −2.23558945544576002364252178924, −0.73809877563537059813746366595, 2.08261707933535427855262056700, 3.11855241090163398539633654517, 3.758072787186427155427310684767, 5.174403024782006709775566738919, 6.25634598710431942247707135932, 6.90778777377355901853612179986, 7.909172374408265471238573750233, 8.76503114415427008555422968954, 9.82836452120804452033307113245, 10.32218332731144546335573976040, 12.23525188714352149247809082381, 12.915963949822642510486559520906, 13.84060649159796197884556333447, 14.32446745969467796297368763687, 15.09712842231607618229407952251, 16.11764569899224127047315187361, 16.64237138447520248395105939139, 17.98376240283345246111066475360, 18.71505166989185055949023268313, 19.28024875238178514970399744369, 20.66604970563883604616851360361, 21.565856903205916156323561976840, 21.88823123279763067944877453774, 23.034175002822587972200707494095, 23.80315751716201468588117787193

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