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Properties

Label 1-1183-1183.500-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.436 - 0.899i$
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  + (−0.992 − 0.120i)2-s + (0.200 + 0.979i)3-s + (0.970 + 0.239i)4-s + (−0.999 − 0.0402i)5-s + (−0.0804 − 0.996i)6-s + (−0.935 − 0.354i)8-s + (−0.919 + 0.391i)9-s + (0.987 + 0.160i)10-s + (0.391 − 0.919i)11-s + (−0.0402 + 0.999i)12-s + (−0.160 − 0.987i)15-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (0.960 − 0.278i)18-s + (0.866 − 0.5i)19-s + (−0.960 − 0.278i)20-s + ⋯
L(s)  = 1  + (−0.992 − 0.120i)2-s + (0.200 + 0.979i)3-s + (0.970 + 0.239i)4-s + (−0.999 − 0.0402i)5-s + (−0.0804 − 0.996i)6-s + (−0.935 − 0.354i)8-s + (−0.919 + 0.391i)9-s + (0.987 + 0.160i)10-s + (0.391 − 0.919i)11-s + (−0.0402 + 0.999i)12-s + (−0.160 − 0.987i)15-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (0.960 − 0.278i)18-s + (0.866 − 0.5i)19-s + (−0.960 − 0.278i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.436 - 0.899i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (500, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.436 - 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3347895769 - 0.2096008503i\)
\(L(\frac12)\) \(\approx\) \(0.3347895769 - 0.2096008503i\)
\(L(1)\) \(\approx\) \(0.5343322754 + 0.08688882359i\)
\(L(1)\) \(\approx\) \(0.5343322754 + 0.08688882359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.992 - 0.120i)T \)
3 \( 1 + (0.200 + 0.979i)T \)
5 \( 1 + (-0.999 - 0.0402i)T \)
11 \( 1 + (0.391 - 0.919i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.919 + 0.391i)T \)
31 \( 1 + (0.903 + 0.428i)T \)
37 \( 1 + (-0.822 + 0.568i)T \)
41 \( 1 + (-0.316 + 0.948i)T \)
43 \( 1 + (-0.428 - 0.903i)T \)
47 \( 1 + (-0.960 - 0.278i)T \)
53 \( 1 + (0.987 - 0.160i)T \)
59 \( 1 + (-0.464 - 0.885i)T \)
61 \( 1 + (0.632 - 0.774i)T \)
67 \( 1 + (-0.960 - 0.278i)T \)
71 \( 1 + (-0.979 + 0.200i)T \)
73 \( 1 + (-0.600 - 0.799i)T \)
79 \( 1 + (0.278 - 0.960i)T \)
83 \( 1 + (0.663 - 0.748i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.534 - 0.845i)T \)
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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

−20.73792747774505486166844696628, −20.37584344450539748328478553031, −19.59646813319641607946291457830, −19.11934810544077369479871356812, −18.15738129259118119589955103978, −17.88079042791770040407803756647, −16.82278595731697591094974514635, −16.0782443683904750484666082451, −15.24083267458148757096820110790, −14.562576489901920099853313730051, −13.61749055724578708577753584653, −12.40691739620748087000007769813, −11.84482690716598495348005710910, −11.404056750908832063808152461773, −10.17880898928656135740949870616, −9.31586528549039388148081846982, −8.49882549011175740533929891183, −7.64151299834834392914171214698, −7.27744961575310255446230486107, −6.47065457511180581197846738872, −5.4079340342727964814799469514, −4.01035752926978575310432351219, −2.92119011382687361275846206442, −2.003608133954882138918724379977, −0.99179715644888034067857067343, 0.25561047486027275452816137876, 1.724964640701781794637833222665, 3.18085025492281832568819259049, 3.5142858269649612182439039963, 4.63686768073803469441360797429, 5.808517221110962497648579245, 6.771071443229967993733983190008, 7.87403997925985463794263163479, 8.483418261414361696368398056698, 9.04668040701899886664483633472, 10.072667606526623387639248381110, 10.719936645069759141764142325550, 11.5645381987124781714790945232, 11.91843335608370874232403579872, 13.293554888208596412454298790072, 14.42375177716911992680632071588, 15.214459217934739077823906585008, 15.81920893195661939699367227689, 16.42647195251087300134204159273, 17.058876125490235968409299851254, 18.01704387593615332907923023580, 19.01046064160034597794145100547, 19.59203083124019269066854803211, 20.150205423269043956767892816123, 20.83759889311080036506032816290

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