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Properties

Label 1-1183-1183.985-r1-0-0
Degree $1$
Conductor $1183$
Sign $-0.221 + 0.975i$
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  + (−0.885 + 0.464i)2-s + (−0.692 − 0.721i)3-s + (0.568 − 0.822i)4-s + (0.987 − 0.160i)5-s + (0.948 + 0.316i)6-s + (−0.120 + 0.992i)8-s + (−0.0402 + 0.999i)9-s + (−0.799 + 0.600i)10-s + (0.0402 + 0.999i)11-s + (−0.987 + 0.160i)12-s + (−0.799 − 0.600i)15-s + (−0.354 − 0.935i)16-s + (−0.120 + 0.992i)17-s + (−0.428 − 0.903i)18-s + (−0.5 + 0.866i)19-s + (0.428 − 0.903i)20-s + ⋯
L(s)  = 1  + (−0.885 + 0.464i)2-s + (−0.692 − 0.721i)3-s + (0.568 − 0.822i)4-s + (0.987 − 0.160i)5-s + (0.948 + 0.316i)6-s + (−0.120 + 0.992i)8-s + (−0.0402 + 0.999i)9-s + (−0.799 + 0.600i)10-s + (0.0402 + 0.999i)11-s + (−0.987 + 0.160i)12-s + (−0.799 − 0.600i)15-s + (−0.354 − 0.935i)16-s + (−0.120 + 0.992i)17-s + (−0.428 − 0.903i)18-s + (−0.5 + 0.866i)19-s + (0.428 − 0.903i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (985, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ -0.221 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6249040468 + 0.7827604664i\)
\(L(\frac12)\) \(\approx\) \(0.6249040468 + 0.7827604664i\)
\(L(1)\) \(\approx\) \(0.6834374252 + 0.1098915371i\)
\(L(1)\) \(\approx\) \(0.6834374252 + 0.1098915371i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.885 + 0.464i)T \)
3 \( 1 + (-0.692 - 0.721i)T \)
5 \( 1 + (0.987 - 0.160i)T \)
11 \( 1 + (0.0402 + 0.999i)T \)
17 \( 1 + (-0.120 + 0.992i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.0402 + 0.999i)T \)
31 \( 1 + (-0.200 - 0.979i)T \)
37 \( 1 + (0.748 - 0.663i)T \)
41 \( 1 + (0.278 - 0.960i)T \)
43 \( 1 + (-0.200 + 0.979i)T \)
47 \( 1 + (0.428 - 0.903i)T \)
53 \( 1 + (0.799 + 0.600i)T \)
59 \( 1 + (-0.354 + 0.935i)T \)
61 \( 1 + (0.919 + 0.391i)T \)
67 \( 1 + (-0.428 + 0.903i)T \)
71 \( 1 + (-0.692 - 0.721i)T \)
73 \( 1 + (-0.845 + 0.534i)T \)
79 \( 1 + (0.428 - 0.903i)T \)
83 \( 1 + (-0.970 - 0.239i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.632 + 0.774i)T \)
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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

−20.92735747271227695602551632127, −20.18030952938233343558409429947, −19.09779196871845382290886166729, −18.41009282729491521734722999640, −17.63587449042714830105425317546, −17.10491838101724502678123747523, −16.42271973027162532694921649126, −15.713877177437076197405301214, −14.77751447919327191412991216892, −13.627230114289876695877480220436, −12.92078130099374775804708481574, −11.81148696967982350091296031491, −11.116256112591932313469075551214, −10.62847834527380534602103056435, −9.67617413708522671540485001580, −9.20058900627339209374072429763, −8.40357557276322889950990063262, −6.999379556951368312290387495492, −6.4050852341958808555222823437, −5.44979330749169518571882251989, −4.48604216333591348463930854923, −3.22188530101876656315584807015, −2.577893247693614500752245417818, −1.178008093168792993448654794031, −0.35023440689029097609246427275, 1.02879913833186157678484475175, 1.766529213872964700489117275917, 2.47240119405791674014751384930, 4.42564596856076629188717992764, 5.507890787470184941348777221879, 5.98317808677960339580088738413, 6.86837569232035832301604412692, 7.492175861226863492546630812177, 8.52068730063005848909823541576, 9.30374518931297119887945988398, 10.337323213261945895732903340390, 10.66491428638207512618151906226, 11.80870893164637487533974085903, 12.706876993228684135040594796522, 13.30569595647441271089864520292, 14.507376355993680049449079399261, 14.9760220994987800120784411541, 16.27513306991911931311616905584, 16.87748695708145305194066271471, 17.42026114549245869773076599280, 18.03380379503982420321779381658, 18.68477072954614817882793506033, 19.46747014006387745926040041000, 20.32236740289642294991699731076, 21.15358004434366054868787308454

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