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Properties

Label 1-443-443.59-r0-0-0
Degree $1$
Conductor $443$
Sign $-0.966 + 0.258i$
Analytic cond. $2.05728$
Root an. cond. $2.05728$
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  + (0.0922 + 0.995i)2-s + (0.445 − 0.895i)3-s + (−0.982 + 0.183i)4-s + (−0.273 − 0.961i)5-s + (0.932 + 0.361i)6-s + (−0.273 + 0.961i)7-s + (−0.273 − 0.961i)8-s + (−0.602 − 0.798i)9-s + (0.932 − 0.361i)10-s + (−0.850 + 0.526i)11-s + (−0.273 + 0.961i)12-s + (−0.273 + 0.961i)13-s + (−0.982 − 0.183i)14-s + (−0.982 − 0.183i)15-s + (0.932 − 0.361i)16-s + (−0.982 + 0.183i)17-s + ⋯
L(s)  = 1  + (0.0922 + 0.995i)2-s + (0.445 − 0.895i)3-s + (−0.982 + 0.183i)4-s + (−0.273 − 0.961i)5-s + (0.932 + 0.361i)6-s + (−0.273 + 0.961i)7-s + (−0.273 − 0.961i)8-s + (−0.602 − 0.798i)9-s + (0.932 − 0.361i)10-s + (−0.850 + 0.526i)11-s + (−0.273 + 0.961i)12-s + (−0.273 + 0.961i)13-s + (−0.982 − 0.183i)14-s + (−0.982 − 0.183i)15-s + (0.932 − 0.361i)16-s + (−0.982 + 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 443 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(443\)
Sign: $-0.966 + 0.258i$
Analytic conductor: \(2.05728\)
Root analytic conductor: \(2.05728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{443} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 443,\ (0:\ ),\ -0.966 + 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04742843354 + 0.3614263675i\)
\(L(\frac12)\) \(\approx\) \(0.04742843354 + 0.3614263675i\)
\(L(1)\) \(\approx\) \(0.7175409473 + 0.2163451068i\)
\(L(1)\) \(\approx\) \(0.7175409473 + 0.2163451068i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad443 \( 1 \)
good2 \( 1 + (0.0922 + 0.995i)T \)
3 \( 1 + (0.445 - 0.895i)T \)
5 \( 1 + (-0.273 - 0.961i)T \)
7 \( 1 + (-0.273 + 0.961i)T \)
11 \( 1 + (-0.850 + 0.526i)T \)
13 \( 1 + (-0.273 + 0.961i)T \)
17 \( 1 + (-0.982 + 0.183i)T \)
19 \( 1 + (0.0922 - 0.995i)T \)
23 \( 1 + (-0.273 + 0.961i)T \)
29 \( 1 + (0.445 + 0.895i)T \)
31 \( 1 + (0.739 + 0.673i)T \)
37 \( 1 + (-0.273 - 0.961i)T \)
41 \( 1 + (-0.273 + 0.961i)T \)
43 \( 1 + (-0.602 + 0.798i)T \)
47 \( 1 + (-0.982 - 0.183i)T \)
53 \( 1 + (0.932 + 0.361i)T \)
59 \( 1 + (-0.982 + 0.183i)T \)
61 \( 1 + (-0.602 - 0.798i)T \)
67 \( 1 + (0.0922 + 0.995i)T \)
71 \( 1 + (-0.850 + 0.526i)T \)
73 \( 1 + (-0.602 - 0.798i)T \)
79 \( 1 + (0.932 + 0.361i)T \)
83 \( 1 + (0.932 + 0.361i)T \)
89 \( 1 + (-0.982 - 0.183i)T \)
97 \( 1 + (-0.602 - 0.798i)T \)
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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

−23.091919830165166859598009654529, −22.64378891179979343976374703164, −21.97910761174434396082395243836, −20.87757528083890431953632568609, −20.3733817541804479703485267668, −19.52099650249014630574030163854, −18.77771596077436917417813217338, −17.79318740423512388170392586824, −16.7584951804637351032185068287, −15.594717632521024849799253508999, −14.8541633631888128874104244077, −13.80211931323880374507992322492, −13.39691731776552120843038801299, −11.95970678532969761420393050285, −10.865135755947050759586093339464, −10.37267986944757518832320337681, −9.892775012176006527308318731069, −8.42165573794220314436585255722, −7.75769395114513345122707929076, −6.153717866643306731499246866395, −4.86032319163646300040272386322, −3.91544021164564131361586774791, −3.127493262140520765679955806759, −2.356162796857120357965412163996, −0.1808170135272862678990097311, 1.66008639871224320702435368639, 2.98384105454794082153744658325, 4.49185447442346049233066453163, 5.31010012355751107187169155446, 6.44394018513844166268527555657, 7.26317445792754082013727646547, 8.27590112149109188958725986606, 8.92126364116235252337946946995, 9.59228232854633925374494810145, 11.61549235718417313163520179918, 12.47845657468890880981448210151, 13.1016027883652319116609620213, 13.841554135416830275058976670413, 15.066039381235345981053024700577, 15.639521749831241713012118849119, 16.51575672509460875615872644251, 17.76106253040522647825735662668, 18.06494638540635810701041199985, 19.318403925704112250853490707281, 19.79502034247733896794328059179, 21.1923226650805762977386242522, 21.8908196415142788042016177449, 23.30089002061551109173715960245, 23.66399102080830634997271606159, 24.63231302517884510340664047934

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