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Properties

Label 1-1183-1183.1055-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.879 - 0.475i$
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  + (0.239 + 0.970i)2-s + (−0.799 + 0.600i)3-s + (−0.885 + 0.464i)4-s + (0.903 + 0.428i)5-s + (−0.774 − 0.632i)6-s + (−0.663 − 0.748i)8-s + (0.278 − 0.960i)9-s + (−0.200 + 0.979i)10-s + (−0.960 + 0.278i)11-s + (0.428 − 0.903i)12-s + (−0.979 + 0.200i)15-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s + (0.999 + 0.0402i)18-s + (−0.866 − 0.5i)19-s + (−0.999 + 0.0402i)20-s + ⋯
L(s)  = 1  + (0.239 + 0.970i)2-s + (−0.799 + 0.600i)3-s + (−0.885 + 0.464i)4-s + (0.903 + 0.428i)5-s + (−0.774 − 0.632i)6-s + (−0.663 − 0.748i)8-s + (0.278 − 0.960i)9-s + (−0.200 + 0.979i)10-s + (−0.960 + 0.278i)11-s + (0.428 − 0.903i)12-s + (−0.979 + 0.200i)15-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s + (0.999 + 0.0402i)18-s + (−0.866 − 0.5i)19-s + (−0.999 + 0.0402i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.879 - 0.475i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.879 - 0.475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2451163300 - 0.06199329499i\)
\(L(\frac12)\) \(\approx\) \(0.2451163300 - 0.06199329499i\)
\(L(1)\) \(\approx\) \(0.5472568900 + 0.4608585038i\)
\(L(1)\) \(\approx\) \(0.5472568900 + 0.4608585038i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.239 + 0.970i)T \)
3 \( 1 + (-0.799 + 0.600i)T \)
5 \( 1 + (0.903 + 0.428i)T \)
11 \( 1 + (-0.960 + 0.278i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (0.278 - 0.960i)T \)
31 \( 1 + (-0.160 + 0.987i)T \)
37 \( 1 + (0.935 - 0.354i)T \)
41 \( 1 + (0.391 - 0.919i)T \)
43 \( 1 + (-0.987 + 0.160i)T \)
47 \( 1 + (-0.999 + 0.0402i)T \)
53 \( 1 + (-0.200 - 0.979i)T \)
59 \( 1 + (0.822 - 0.568i)T \)
61 \( 1 + (-0.948 + 0.316i)T \)
67 \( 1 + (-0.999 + 0.0402i)T \)
71 \( 1 + (-0.600 - 0.799i)T \)
73 \( 1 + (-0.721 - 0.692i)T \)
79 \( 1 + (-0.0402 - 0.999i)T \)
83 \( 1 + (0.992 + 0.120i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.0804 - 0.996i)T \)
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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

−21.48663254008324814984100548707, −20.5111413343169393318523649969, −19.888438013677171267779782264363, −18.80944825176321625173053443966, −18.19254479627363562914332887948, −17.80974049814197613122559683735, −16.81358251297764242977321499006, −16.13958984890894057032490080737, −14.848927150413487705753249898, −13.83225295007001669301181020736, −13.2391764787701572740432305414, −12.791399132871607869707409866515, −11.92128966240939553717943256263, −11.10995870847606754707058936280, −10.38251570064582086645638108991, −9.73571368217307704024908282051, −8.65733002340298299085924685765, −7.84871563962939892773635394024, −6.44069365922897157956876882168, −5.82362488256298636637600989238, −5.0133235864218741967908672407, −4.32663796133897449382963920102, −2.7601632356182342787106137696, −2.07749625023372915037110441625, −1.163658268696987477452805887959, 0.1080300262536686367785051101, 2.00725783953807946519951653085, 3.25261628090130886793541073636, 4.38474445154251709428174049215, 4.99398973459860406775433049280, 5.996616434341153558630450217636, 6.34528326746175815708220679042, 7.29836842388355488023976240412, 8.41205402961175162506718735614, 9.2908872920560680359788910886, 10.15745696095272304665829432085, 10.653215209344044600888618725825, 11.80888951113743787795892616218, 12.9013926441354129760627744262, 13.29841355335461767447517827264, 14.43210174076238516952537415455, 15.06732454187968061751262321525, 15.7656216237700524514817748661, 16.47963123829386965291678209212, 17.38167201657296156337448026983, 17.8128966935315503923194173757, 18.32216428313006319303728741962, 19.50411461840414252582072099133, 20.92122993198549599813099419460, 21.366856559122032976574074255707

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