↑

Properties

Label 1-1183-1183.1060-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.998 - 0.0467i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.464 + 0.885i)2-s + (−0.278 + 0.960i)3-s + (−0.568 + 0.822i)4-s + (−0.774 + 0.632i)5-s + (−0.979 + 0.200i)6-s + (−0.992 − 0.120i)8-s + (−0.845 − 0.534i)9-s + (−0.919 − 0.391i)10-s + (0.534 + 0.845i)11-s + (−0.632 − 0.774i)12-s + (−0.391 − 0.919i)15-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + (0.0804 − 0.996i)18-s + (−0.866 + 0.5i)19-s + (−0.0804 − 0.996i)20-s + ⋯
L(s)  = 1  + (0.464 + 0.885i)2-s + (−0.278 + 0.960i)3-s + (−0.568 + 0.822i)4-s + (−0.774 + 0.632i)5-s + (−0.979 + 0.200i)6-s + (−0.992 − 0.120i)8-s + (−0.845 − 0.534i)9-s + (−0.919 − 0.391i)10-s + (0.534 + 0.845i)11-s + (−0.632 − 0.774i)12-s + (−0.391 − 0.919i)15-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + (0.0804 − 0.996i)18-s + (−0.866 + 0.5i)19-s + (−0.0804 − 0.996i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.998 - 0.0467i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1060, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.998 - 0.0467i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1851699636 + 0.004329116365i\)
\(L(\frac12)\) \(\approx\) \(0.1851699636 + 0.004329116365i\)
\(L(1)\) \(\approx\) \(0.4775508311 + 0.5911383119i\)
\(L(1)\) \(\approx\) \(0.4775508311 + 0.5911383119i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.464 + 0.885i)T \)
3 \( 1 + (-0.278 + 0.960i)T \)
5 \( 1 + (-0.774 + 0.632i)T \)
11 \( 1 + (0.534 + 0.845i)T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.845 - 0.534i)T \)
31 \( 1 + (0.316 - 0.948i)T \)
37 \( 1 + (0.663 + 0.748i)T \)
41 \( 1 + (0.721 - 0.692i)T \)
43 \( 1 + (-0.948 + 0.316i)T \)
47 \( 1 + (-0.0804 - 0.996i)T \)
53 \( 1 + (-0.919 + 0.391i)T \)
59 \( 1 + (-0.935 - 0.354i)T \)
61 \( 1 + (-0.799 + 0.600i)T \)
67 \( 1 + (-0.0804 - 0.996i)T \)
71 \( 1 + (0.960 + 0.278i)T \)
73 \( 1 + (0.999 - 0.0402i)T \)
79 \( 1 + (-0.996 + 0.0804i)T \)
83 \( 1 + (-0.239 + 0.970i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.160 - 0.987i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.35844039670825349375145927564, −20.11579860575998432139046711454, −19.73821888663285222564373695324, −19.14116473922432615677314591665, −18.45734085908912005177968100245, −17.48804372572423264358203088417, −16.75257308096274153334605239840, −15.80843412510054938292771564087, −14.7042974644549744135716546350, −14.08636773718301379123853034718, −13.03794914564246904730527955431, −12.696636478180223498506276286779, −11.89544593190930240149083581216, −11.22597027189684291418544681585, −10.62507328407836520245770637386, −9.22708914804374958694214498877, −8.525779030612140881768199197862, −7.80763300718147137606456854507, −6.48898957285599978790178389514, −5.87723867631876735747827792016, −4.857659024389740644469067392011, −3.95235134974772090857897632811, −3.09729547992689043316435272062, −1.86641839642764485330327801979, −1.09960094484100707732200259591, 0.07386837310985505032631735395, 2.45068721103526708271273833383, 3.53907907781832030695617790914, 4.17693910142359354338696614721, 4.7900459903865156341822834825, 5.946912876915743678943818737429, 6.59547650319474299314746080278, 7.578411374564987947746939699169, 8.2743720158884379330024377065, 9.39147733864512150447773044317, 9.97705080288429884587705242091, 11.16404339062899003079641708632, 11.84446726084892146166695083089, 12.49436659361538035924181002239, 13.80988489933435535167813787896, 14.49045375325885895673398989492, 15.23551354437793288663203500728, 15.51845013756050369817021061496, 16.58069113702918944655149465121, 16.97951211564922414358408596228, 18.01829640195244212909643770826, 18.65579860475645257966640173801, 19.896914430646015772117373234953, 20.57268444033628840529955988638, 21.44997182903689155346184859152

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