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Properties

Label 1-1183-1183.475-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.585 + 0.810i$
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  + (−0.979 − 0.200i)2-s + (−0.987 + 0.160i)3-s + (0.919 + 0.391i)4-s + (−0.239 + 0.970i)5-s + (0.999 + 0.0402i)6-s + (−0.822 − 0.568i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (0.316 − 0.948i)11-s + (−0.970 − 0.239i)12-s + (0.0804 − 0.996i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (−0.992 + 0.120i)18-s + (0.866 + 0.5i)19-s + (−0.600 + 0.799i)20-s + ⋯
L(s)  = 1  + (−0.979 − 0.200i)2-s + (−0.987 + 0.160i)3-s + (0.919 + 0.391i)4-s + (−0.239 + 0.970i)5-s + (0.999 + 0.0402i)6-s + (−0.822 − 0.568i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (0.316 − 0.948i)11-s + (−0.970 − 0.239i)12-s + (0.0804 − 0.996i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (−0.992 + 0.120i)18-s + (0.866 + 0.5i)19-s + (−0.600 + 0.799i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.585 + 0.810i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.585 + 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5874205771 + 0.3002232922i\)
\(L(\frac12)\) \(\approx\) \(0.5874205771 + 0.3002232922i\)
\(L(1)\) \(\approx\) \(0.5477913526 + 0.09758015320i\)
\(L(1)\) \(\approx\) \(0.5477913526 + 0.09758015320i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.979 - 0.200i)T \)
3 \( 1 + (-0.987 + 0.160i)T \)
5 \( 1 + (-0.239 + 0.970i)T \)
11 \( 1 + (0.316 - 0.948i)T \)
17 \( 1 + (0.428 + 0.903i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.200 + 0.979i)T \)
31 \( 1 + (-0.464 - 0.885i)T \)
37 \( 1 + (-0.534 + 0.845i)T \)
41 \( 1 + (-0.160 - 0.987i)T \)
43 \( 1 + (0.845 - 0.534i)T \)
47 \( 1 + (0.992 + 0.120i)T \)
53 \( 1 + (0.568 - 0.822i)T \)
59 \( 1 + (0.721 + 0.692i)T \)
61 \( 1 + (0.996 - 0.0804i)T \)
67 \( 1 + (-0.391 - 0.919i)T \)
71 \( 1 + (0.774 - 0.632i)T \)
73 \( 1 + (-0.663 - 0.748i)T \)
79 \( 1 + (0.120 - 0.992i)T \)
83 \( 1 + (-0.935 + 0.354i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.960 - 0.278i)T \)
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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

−20.79919467885062131837096604745, −20.335465501472924026762055333180, −19.45145407032081351951090532158, −18.651182489868665181231765833252, −17.81131277713322001524575207858, −17.35925369035606554347594777662, −16.53692908185894397271382814885, −16.02889201166030161875857453806, −15.35246963052005174251351320351, −14.23420643834006198921738254715, −12.9844922014629708518983901550, −12.22831881369213274595960610145, −11.6923673155673854368751407847, −10.897090860707399040756500421907, −9.83557078421347077675513737577, −9.380799247375521415008531665248, −8.36871325894868289322787989660, −7.32283541949661883047462003705, −6.96728274922495595369076464018, −5.71625135639240393495600386971, −5.1063644700904112518561114294, −4.16848768902785948087818018653, −2.53265959779452718860178493398, −1.36864523248291073813420417225, −0.63756995840770726508060791275, 0.85005828919585552736303427735, 1.90461266521927155328097508538, 3.34056874877236121956627893968, 3.76867869738326608915418053673, 5.52371191340226287664038910524, 6.082691203259926348723636949, 7.04936852351062297183128754232, 7.59391403924655173950328827216, 8.70637364013146099683422153327, 9.677417156074563473499070982478, 10.46410761699475233789915619465, 10.97451206750721850171824167404, 11.688430267624490798675173876841, 12.283649061434362354352978111979, 13.454665656425263257621617551383, 14.62363968821651307681756815582, 15.415763640982407192414650696487, 16.13462355900901784595085010436, 16.86657982023481946264177954368, 17.503427368414869006233497824718, 18.33040880589329931587555216072, 18.9050956668612059468976296888, 19.42189411591280345705313987024, 20.59172961636109629510694757302, 21.38597897384088871145354180762

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