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Properties

Label 1-1183-1183.1069-r1-0-0
Degree $1$
Conductor $1183$
Sign $-0.853 + 0.521i$
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  + (−0.748 + 0.663i)2-s + (0.632 − 0.774i)3-s + (0.120 − 0.992i)4-s + (−0.692 − 0.721i)5-s + (0.0402 + 0.999i)6-s + (0.568 + 0.822i)8-s + (−0.200 − 0.979i)9-s + (0.996 + 0.0804i)10-s + (−0.200 + 0.979i)11-s + (−0.692 − 0.721i)12-s + (−0.996 + 0.0804i)15-s + (−0.970 − 0.239i)16-s + (−0.568 − 0.822i)17-s + (0.799 + 0.600i)18-s + (0.5 − 0.866i)19-s + (−0.799 + 0.600i)20-s + ⋯
L(s)  = 1  + (−0.748 + 0.663i)2-s + (0.632 − 0.774i)3-s + (0.120 − 0.992i)4-s + (−0.692 − 0.721i)5-s + (0.0402 + 0.999i)6-s + (0.568 + 0.822i)8-s + (−0.200 − 0.979i)9-s + (0.996 + 0.0804i)10-s + (−0.200 + 0.979i)11-s + (−0.692 − 0.721i)12-s + (−0.996 + 0.0804i)15-s + (−0.970 − 0.239i)16-s + (−0.568 − 0.822i)17-s + (0.799 + 0.600i)18-s + (0.5 − 0.866i)19-s + (−0.799 + 0.600i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.853 + 0.521i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ -0.853 + 0.521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1088254720 - 0.3871035110i\)
\(L(\frac12)\) \(\approx\) \(-0.1088254720 - 0.3871035110i\)
\(L(1)\) \(\approx\) \(0.6840573723 - 0.1910175671i\)
\(L(1)\) \(\approx\) \(0.6840573723 - 0.1910175671i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.748 + 0.663i)T \)
3 \( 1 + (0.632 - 0.774i)T \)
5 \( 1 + (-0.692 - 0.721i)T \)
11 \( 1 + (-0.200 + 0.979i)T \)
17 \( 1 + (-0.568 - 0.822i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.200 - 0.979i)T \)
31 \( 1 + (0.845 - 0.534i)T \)
37 \( 1 + (0.885 + 0.464i)T \)
41 \( 1 + (-0.987 - 0.160i)T \)
43 \( 1 + (-0.845 - 0.534i)T \)
47 \( 1 + (-0.799 + 0.600i)T \)
53 \( 1 + (-0.996 + 0.0804i)T \)
59 \( 1 + (0.970 - 0.239i)T \)
61 \( 1 + (-0.428 - 0.903i)T \)
67 \( 1 + (0.799 - 0.600i)T \)
71 \( 1 + (-0.632 + 0.774i)T \)
73 \( 1 + (-0.948 + 0.316i)T \)
79 \( 1 + (0.799 - 0.600i)T \)
83 \( 1 + (0.354 - 0.935i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.278 - 0.960i)T \)
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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

−21.402000499886017606452378874, −20.63159017180563567734548909148, −19.78465669644197893403085878510, −19.30971961794833445785178175483, −18.65646358408909932474766796109, −17.86122418724044233488331963745, −16.64037488145677237937891762448, −16.264445233624185352327976649415, −15.347503787620957467685724857812, −14.64469070846507698804614322013, −13.6872749140835900304946805229, −12.86565453036732841274735856007, −11.71120502469429698633980989348, −11.051869453132550061770155970551, −10.48859705814356270300601019003, −9.7571107100336181304921278143, −8.64723904502902911257659755587, −8.29121135311439280074829099949, −7.41434604190168232482798332893, −6.403198902145083147136350449777, −4.96529786288835511650628830394, −3.8693151596861030706134373037, −3.30608614749912944161200590369, −2.6424420451428558507897258134, −1.37721509300302188588287011071, 0.113954363641171386589492493017, 0.92369984128176771389527543354, 1.95416956082411284964628924953, 2.97379180897081619096143531832, 4.46902413916756941681051956082, 5.1072589693918470964789171011, 6.446761626819307165994911448169, 7.148504278178325053751297938892, 7.772415767437179961866701294898, 8.48765541303386427908838313779, 9.29489520534036357360496751669, 9.82610441528710946848511841543, 11.32279407749941140268224496797, 11.813995965091664688165080373350, 13.01991945617400292285827402680, 13.48360018795230402417296946173, 14.57670880658515323749604458281, 15.41840543280233921763680708699, 15.677258783560284310215005586124, 16.9396188987013991819572767576, 17.47889201821808370139452048288, 18.31016116525475861447722854083, 19.00820571230015056211130646460, 19.69611633393968201234246119681, 20.40966782985382453281090794526

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