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Properties

Label 1-1183-1183.920-r1-0-0
Degree $1$
Conductor $1183$
Sign $-0.733 + 0.679i$
Analytic cond. $127.131$
Root an. cond. $127.131$
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  + (0.0402 + 0.999i)2-s + (0.970 + 0.239i)3-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (−0.200 + 0.979i)6-s + (−0.120 − 0.992i)8-s + (0.885 + 0.464i)9-s + (−0.120 + 0.992i)10-s + (−0.885 + 0.464i)11-s + (−0.987 − 0.160i)12-s + (0.919 + 0.391i)15-s + (0.987 − 0.160i)16-s + (0.919 + 0.391i)17-s + (−0.428 + 0.903i)18-s + 19-s + (−0.996 − 0.0804i)20-s + ⋯
L(s)  = 1  + (0.0402 + 0.999i)2-s + (0.970 + 0.239i)3-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (−0.200 + 0.979i)6-s + (−0.120 − 0.992i)8-s + (0.885 + 0.464i)9-s + (−0.120 + 0.992i)10-s + (−0.885 + 0.464i)11-s + (−0.987 − 0.160i)12-s + (0.919 + 0.391i)15-s + (0.987 − 0.160i)16-s + (0.919 + 0.391i)17-s + (−0.428 + 0.903i)18-s + 19-s + (−0.996 − 0.0804i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.733 + 0.679i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (920, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ -0.733 + 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.357743129 + 3.464827272i\)
\(L(\frac12)\) \(\approx\) \(1.357743129 + 3.464827272i\)
\(L(1)\) \(\approx\) \(1.300602599 + 1.100017322i\)
\(L(1)\) \(\approx\) \(1.300602599 + 1.100017322i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.0402 + 0.999i)T \)
3 \( 1 + (0.970 + 0.239i)T \)
5 \( 1 + (0.987 + 0.160i)T \)
11 \( 1 + (-0.885 + 0.464i)T \)
17 \( 1 + (0.919 + 0.391i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.845 + 0.534i)T \)
31 \( 1 + (-0.200 + 0.979i)T \)
37 \( 1 + (0.200 - 0.979i)T \)
41 \( 1 + (0.692 - 0.721i)T \)
43 \( 1 + (0.948 + 0.316i)T \)
47 \( 1 + (0.428 + 0.903i)T \)
53 \( 1 + (0.799 - 0.600i)T \)
59 \( 1 + (0.987 + 0.160i)T \)
61 \( 1 + (-0.120 + 0.992i)T \)
67 \( 1 + (-0.568 + 0.822i)T \)
71 \( 1 + (-0.278 - 0.960i)T \)
73 \( 1 + (-0.845 - 0.534i)T \)
79 \( 1 + (0.428 + 0.903i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.987 - 0.160i)T \)
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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

−20.53223606054689064280586482975, −20.33633750135465541035274358995, −19.15866848852019316811112160766, −18.54897275510973133751311615922, −18.078634454399025445993213680721, −17.12017541981690613448423274461, −16.10087235081174122823230370558, −15.00870718762966141869378767795, −14.16771449322505801421071877518, −13.53085161848189951601637558437, −13.18097095067431544079707759211, −12.23212617969327438124585781623, −11.337724680625213116577468674275, −10.19159072901029918947290626501, −9.72101599614781948661190851151, −9.09146097631525256945762829899, −8.086634774431018021521980529453, −7.44685319113482132213660311051, −5.87480295146479236701885409601, −5.28554285359365708891508397520, −4.076835567024528239578132385556, −3.109155406644257700569069384123, −2.49117684799131253072606030225, −1.57688536091925516562991056401, −0.68424927131471835825804183372, 1.12487292915127036230762079218, 2.323195819658800373222181298731, 3.25541110599115486551232501752, 4.281655325307260322950807645714, 5.28650348268607433229519463706, 5.86522141840887434312173163503, 7.174106170790175318420540210809, 7.572500145002629038672174421595, 8.6188403506911358133897540016, 9.28599540618508623420989151580, 10.065583646902145313407041226941, 10.58330195396469237233573166403, 12.47783336454142930042378563273, 12.96208205865899178168407073346, 13.856536597946271970941453928695, 14.41534310101098994189692068698, 14.92069944653090368866407672935, 16.04483121114661084195404038451, 16.35478131296770694041326332339, 17.57305019252177749359880518058, 18.16734404329804960214662290089, 18.76491305356864989120619687147, 19.74458970139174406648162780973, 20.89164640070193306392315774209, 21.1508696083486140408457757142

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