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Properties

Label 1-547-547.32-r1-0-0
Degree $1$
Conductor $547$
Sign $-0.246 - 0.969i$
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  + (0.998 + 0.0575i)2-s + (0.900 − 0.433i)3-s + (0.993 + 0.114i)4-s + (−0.641 − 0.767i)5-s + (0.924 − 0.381i)6-s + (0.937 − 0.349i)7-s + (0.985 + 0.171i)8-s + (0.623 − 0.781i)9-s + (−0.596 − 0.802i)10-s + (−0.632 − 0.774i)11-s + (0.944 − 0.327i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (−0.910 − 0.413i)15-s + (0.973 + 0.228i)16-s + (−0.740 + 0.671i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.0575i)2-s + (0.900 − 0.433i)3-s + (0.993 + 0.114i)4-s + (−0.641 − 0.767i)5-s + (0.924 − 0.381i)6-s + (0.937 − 0.349i)7-s + (0.985 + 0.171i)8-s + (0.623 − 0.781i)9-s + (−0.596 − 0.802i)10-s + (−0.632 − 0.774i)11-s + (0.944 − 0.327i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (−0.910 − 0.413i)15-s + (0.973 + 0.228i)16-s + (−0.740 + 0.671i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.246 - 0.969i$
Analytic conductor: \(58.7833\)
Root analytic conductor: \(58.7833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (1:\ ),\ -0.246 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.239974011 - 4.169172190i\)
\(L(\frac12)\) \(\approx\) \(3.239974011 - 4.169172190i\)
\(L(1)\) \(\approx\) \(2.322285145 - 1.068208674i\)
\(L(1)\) \(\approx\) \(2.322285145 - 1.068208674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.998 + 0.0575i)T \)
3 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (-0.641 - 0.767i)T \)
7 \( 1 + (0.937 - 0.349i)T \)
11 \( 1 + (-0.632 - 0.774i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (-0.740 + 0.671i)T \)
19 \( 1 + (0.785 + 0.618i)T \)
23 \( 1 + (-0.407 - 0.913i)T \)
29 \( 1 + (0.851 - 0.524i)T \)
31 \( 1 + (0.154 - 0.987i)T \)
37 \( 1 + (-0.968 + 0.250i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.469 + 0.882i)T \)
47 \( 1 + (-0.919 + 0.391i)T \)
53 \( 1 + (0.0976 + 0.995i)T \)
59 \( 1 + (-0.428 - 0.903i)T \)
61 \( 1 + (0.577 + 0.816i)T \)
67 \( 1 + (0.756 + 0.654i)T \)
71 \( 1 + (0.558 - 0.829i)T \)
73 \( 1 + (-0.910 + 0.413i)T \)
79 \( 1 + (0.999 - 0.0345i)T \)
83 \( 1 + (-0.948 + 0.316i)T \)
89 \( 1 + (0.418 + 0.908i)T \)
97 \( 1 + (-0.614 - 0.788i)T \)
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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

−23.41573265195053553158204710911, −22.42439175724321534091517439624, −21.68108466617479244008817900038, −21.12625500885048817848779236923, −19.99493913723277620812029461710, −19.73740628145480525612628891306, −18.53875042155869048426963751630, −17.62968355791614988919422575118, −15.9488890429367015091216756404, −15.65184262902370671850827298058, −14.767591460101891058637064874465, −14.231422226738236746692619186439, −13.46744393519537555927013272806, −12.18298492339166368552588007327, −11.49991488789595236714966156875, −10.60880835557566623489187952422, −9.67431703371297784391904546429, −8.35400142928072668665363516657, −7.37967626171699172069984276145, −6.92282108186817052344149291391, −5.03320280180007540506708726983, −4.70578586392114270943485141049, −3.51132172523020494205200206497, −2.600122157273149019800618761500, −1.8656578509010596167024344716, 0.75926665293657754450429231559, 1.94370505766938410992739731768, 3.016791713043553388778405866830, 4.0582473942712540596460384577, 4.80490256769812832457998378654, 5.90186728440883348632424635472, 7.226230038331338203568243024114, 8.07714750962536433957714229295, 8.3630971500925246880774474684, 10.02892988663864961703109157099, 11.09970801018270009558717981549, 12.07312088833383917934684672670, 12.74960510320078407488510440637, 13.57982985101069652648534785983, 14.295135540002336124843686446840, 15.177798047459257752250716700517, 15.80593213691457723455151776658, 16.84130062934900454281684984997, 17.865383730332641371361817555894, 19.11484992628296965796253705800, 19.82736958444204901318010166739, 20.621630081817128571982241195544, 20.94611653768742548225160403580, 22.05628227049069850091508058375, 23.169019859406905468753683903268

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