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Properties

Label 1-1183-1183.471-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.790 - 0.611i$
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  + (0.120 − 0.992i)2-s + (0.948 + 0.316i)3-s + (−0.970 − 0.239i)4-s + (−0.845 − 0.534i)5-s + (0.428 − 0.903i)6-s + (−0.354 + 0.935i)8-s + (0.799 + 0.600i)9-s + (−0.632 + 0.774i)10-s + (0.799 − 0.600i)11-s + (−0.845 − 0.534i)12-s + (−0.632 − 0.774i)15-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (0.692 − 0.721i)18-s + (−0.5 + 0.866i)19-s + (0.692 + 0.721i)20-s + ⋯
L(s)  = 1  + (0.120 − 0.992i)2-s + (0.948 + 0.316i)3-s + (−0.970 − 0.239i)4-s + (−0.845 − 0.534i)5-s + (0.428 − 0.903i)6-s + (−0.354 + 0.935i)8-s + (0.799 + 0.600i)9-s + (−0.632 + 0.774i)10-s + (0.799 − 0.600i)11-s + (−0.845 − 0.534i)12-s + (−0.632 − 0.774i)15-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (0.692 − 0.721i)18-s + (−0.5 + 0.866i)19-s + (0.692 + 0.721i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.790 - 0.611i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.790 - 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.692315456 - 0.5782513220i\)
\(L(\frac12)\) \(\approx\) \(1.692315456 - 0.5782513220i\)
\(L(1)\) \(\approx\) \(1.196640754 - 0.4632558911i\)
\(L(1)\) \(\approx\) \(1.196640754 - 0.4632558911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.120 - 0.992i)T \)
3 \( 1 + (0.948 + 0.316i)T \)
5 \( 1 + (-0.845 - 0.534i)T \)
11 \( 1 + (0.799 - 0.600i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.799 + 0.600i)T \)
31 \( 1 + (-0.996 + 0.0804i)T \)
37 \( 1 + (0.568 + 0.822i)T \)
41 \( 1 + (-0.200 - 0.979i)T \)
43 \( 1 + (-0.996 - 0.0804i)T \)
47 \( 1 + (0.692 + 0.721i)T \)
53 \( 1 + (-0.632 - 0.774i)T \)
59 \( 1 + (0.885 - 0.464i)T \)
61 \( 1 + (0.987 + 0.160i)T \)
67 \( 1 + (0.692 + 0.721i)T \)
71 \( 1 + (0.948 + 0.316i)T \)
73 \( 1 + (-0.919 - 0.391i)T \)
79 \( 1 + (0.692 + 0.721i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.0402 - 0.999i)T \)
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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

−21.495540830441238985281448564083, −20.28826976838988290203479163958, −19.68048679411462090615836047854, −18.94289812953951307979239985746, −18.234487932515237337450009664656, −17.53367144550745678900555619469, −16.492342538132769488356607924552, −15.62743625886088371022464113986, −15.03145727400496273470611930154, −14.56906314328166162465469688273, −13.72416624820626875149219301786, −12.95893891045951725733741282228, −12.138324300592067995213040510952, −11.18700328098295915321252144623, −9.8791258597576893615045215884, −9.12287424466767977754075235079, −8.47249790231313092748107626975, −7.52863598232251533032950349527, −6.98418470953008220562558134454, −6.44846076971784042932708152360, −4.879247934518747397207474378198, −4.18472208736016091713921623418, −3.35089760118194167096157991321, −2.394749673170205345442036628115, −0.77124994571425067886196039480, 1.073366857697977976384152254608, 1.945892540891383233545010517649, 3.213400299609662563723226029039, 3.75493423734814689807827538641, 4.43328535169712920353403920552, 5.37861666158334860356643744452, 6.766327920264460517256230466257, 8.05347590657242199124267854437, 8.59402392838638651404060042974, 9.11985682794488786463362336499, 10.148279680236113099774570412770, 10.92951752457125156552594942518, 11.70139574806954209402467273146, 12.710107951009373424336335691192, 13.07423635377129832911283287104, 14.231655991046526577633251113809, 14.69533330659383684171119897704, 15.54764210501549007000223920408, 16.552024010465649451238085458830, 17.24454381887885767587199865905, 18.572093894044604173174334018968, 19.18802583297165617760932432314, 19.6105413103313350973122463155, 20.374538386074717188192248221707, 20.92112490026578755676440239950

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