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Properties

Label 1-1183-1183.1171-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.996 - 0.0883i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
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  + (0.987 − 0.160i)2-s + (−0.996 + 0.0804i)3-s + (0.948 − 0.316i)4-s + (−0.919 + 0.391i)5-s + (−0.970 + 0.239i)6-s + (0.885 − 0.464i)8-s + (0.987 − 0.160i)9-s + (−0.845 + 0.534i)10-s + (0.987 + 0.160i)11-s + (−0.919 + 0.391i)12-s + (0.885 − 0.464i)15-s + (0.799 − 0.600i)16-s + (−0.0402 + 0.999i)17-s + (0.948 − 0.316i)18-s + (−0.5 − 0.866i)19-s + (−0.748 + 0.663i)20-s + ⋯
L(s)  = 1  + (0.987 − 0.160i)2-s + (−0.996 + 0.0804i)3-s + (0.948 − 0.316i)4-s + (−0.919 + 0.391i)5-s + (−0.970 + 0.239i)6-s + (0.885 − 0.464i)8-s + (0.987 − 0.160i)9-s + (−0.845 + 0.534i)10-s + (0.987 + 0.160i)11-s + (−0.919 + 0.391i)12-s + (0.885 − 0.464i)15-s + (0.799 − 0.600i)16-s + (−0.0402 + 0.999i)17-s + (0.948 − 0.316i)18-s + (−0.5 − 0.866i)19-s + (−0.748 + 0.663i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.996 - 0.0883i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.996 - 0.0883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.922724973 - 0.08507591478i\)
\(L(\frac12)\) \(\approx\) \(1.922724973 - 0.08507591478i\)
\(L(1)\) \(\approx\) \(1.371889268 - 0.05425319903i\)
\(L(1)\) \(\approx\) \(1.371889268 - 0.05425319903i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.987 - 0.160i)T \)
3 \( 1 + (-0.996 + 0.0804i)T \)
5 \( 1 + (-0.919 + 0.391i)T \)
11 \( 1 + (0.987 + 0.160i)T \)
17 \( 1 + (-0.0402 + 0.999i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.354 + 0.935i)T \)
31 \( 1 + (0.278 - 0.960i)T \)
37 \( 1 + (0.692 + 0.721i)T \)
41 \( 1 + (0.568 + 0.822i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (0.948 + 0.316i)T \)
53 \( 1 + (-0.0402 + 0.999i)T \)
59 \( 1 + (0.799 + 0.600i)T \)
61 \( 1 + (-0.0402 - 0.999i)T \)
67 \( 1 + (-0.200 - 0.979i)T \)
71 \( 1 + (0.568 + 0.822i)T \)
73 \( 1 + (0.987 + 0.160i)T \)
79 \( 1 + (0.948 + 0.316i)T \)
83 \( 1 + (0.568 - 0.822i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.120 + 0.992i)T \)
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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

−21.33137284153861488205111898187, −20.71011585688680338426881090362, −19.684914948005481532236365341476, −19.17768841296193390837059957163, −18.010423064522557826605579023440, −17.05841590789222822267429695429, −16.460369030929723132812022666561, −15.89547135489043699081131503781, −15.142657912020284500723474790553, −14.2247863377905879223970971480, −13.33296711344277230987197467128, −12.45719788698182952191394116116, −11.84455870849903116182064076991, −11.469429409319825185497912571125, −10.56367187580869125574820620131, −9.4391807756470394175154140504, −8.168972713073889812310392608654, −7.3601723058117924385344287355, −6.64436190710199577341255126424, −5.75926586792340349683617707939, −4.99800377732575595052810836118, −4.08685808202762171655893285958, −3.60977320656725170299576937965, −2.05292853754672231563085201385, −0.89930824683415388551312330311, 0.89838807327520818448766672166, 2.119407271888327841554271100159, 3.394457493411819973535289700671, 4.2507951735023821681075393461, 4.65454415395601538398823779479, 5.96066788574661227782994463444, 6.5313851158411813539181544000, 7.19781553988687423172722876708, 8.254253565559561787991835567693, 9.63340823168508296501224014877, 10.672585644269115603465927268421, 11.10706604003139506832222202821, 11.88349545382428858839826693008, 12.45467655978316207614659568473, 13.1944240109156184004988961523, 14.38205465937612545963589902403, 15.07867252631938662854855312361, 15.57586156086641264726691964414, 16.62369530435793360017635147486, 16.99561964716194852668135148667, 18.23847395814881050440749355003, 19.0334585234028337875834194079, 19.79907238759230087020442376201, 20.43866245932853102133172580008, 21.68941101262956953869253336578

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