↑

Properties

Label 1-1183-1183.1116-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.991 - 0.130i$
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  + (0.903 + 0.428i)2-s + (−0.885 − 0.464i)3-s + (0.632 + 0.774i)4-s + (−0.979 + 0.200i)5-s + (−0.600 − 0.799i)6-s + (0.239 + 0.970i)8-s + (0.568 + 0.822i)9-s + (−0.970 − 0.239i)10-s + (−0.822 − 0.568i)11-s + (−0.200 − 0.979i)12-s + (0.960 + 0.278i)15-s + (−0.200 + 0.979i)16-s + (0.278 − 0.960i)17-s + (0.160 + 0.987i)18-s − i·19-s + (−0.774 − 0.632i)20-s + ⋯
L(s)  = 1  + (0.903 + 0.428i)2-s + (−0.885 − 0.464i)3-s + (0.632 + 0.774i)4-s + (−0.979 + 0.200i)5-s + (−0.600 − 0.799i)6-s + (0.239 + 0.970i)8-s + (0.568 + 0.822i)9-s + (−0.970 − 0.239i)10-s + (−0.822 − 0.568i)11-s + (−0.200 − 0.979i)12-s + (0.960 + 0.278i)15-s + (−0.200 + 0.979i)16-s + (0.278 − 0.960i)17-s + (0.160 + 0.987i)18-s − i·19-s + (−0.774 − 0.632i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.991 - 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.375516418 - 0.08981211586i\)
\(L(\frac12)\) \(\approx\) \(1.375516418 - 0.08981211586i\)
\(L(1)\) \(\approx\) \(1.104402343 + 0.1280926073i\)
\(L(1)\) \(\approx\) \(1.104402343 + 0.1280926073i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.903 + 0.428i)T \)
3 \( 1 + (-0.885 - 0.464i)T \)
5 \( 1 + (-0.979 + 0.200i)T \)
11 \( 1 + (-0.822 - 0.568i)T \)
17 \( 1 + (0.278 - 0.960i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.996 + 0.0804i)T \)
31 \( 1 + (-0.600 - 0.799i)T \)
37 \( 1 + (-0.600 - 0.799i)T \)
41 \( 1 + (0.534 + 0.845i)T \)
43 \( 1 + (0.919 - 0.391i)T \)
47 \( 1 + (-0.160 + 0.987i)T \)
53 \( 1 + (0.692 + 0.721i)T \)
59 \( 1 + (0.979 - 0.200i)T \)
61 \( 1 + (0.970 + 0.239i)T \)
67 \( 1 + (0.935 - 0.354i)T \)
71 \( 1 + (0.999 - 0.0402i)T \)
73 \( 1 + (0.0804 - 0.996i)T \)
79 \( 1 + (0.987 + 0.160i)T \)
83 \( 1 + (-0.464 - 0.885i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.979 + 0.200i)T \)
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

−21.07532021262463639204388440121, −20.80508140871919901622493409889, −19.93437908201686564800528475854, −18.94766230415614115854996337849, −18.41240416223094573125403200060, −17.17507482279521476480684963666, −16.35216433927672609394649550883, −15.79476827217444794697761828987, −14.99570512861768904116342826114, −14.529827875348691932297278937820, −13.01788158110429854572814836740, −12.55782236042247755002056517480, −11.97485955263920190988537430427, −11.04381723642968495684033192773, −10.529160614670951177907144383702, −9.78455713059021762585999012703, −8.47962759821916605184803770311, −7.3699239514289827222499664916, −6.61480006265936201661159436344, −5.491103772984250869506289296384, −5.0270681550889171881673157808, −3.96820953957922985894097456638, −3.58579822525061775488974304943, −2.15295006301590055432814562040, −0.89679173394737796773040124421, 0.60287481142231709632136354540, 2.27582200190084138867738111391, 3.20892914619284583909287945090, 4.20588228424566741136281608341, 5.14318446276942611405942685492, 5.664716660381799244938507356936, 6.788104333195626347021208590707, 7.453641398342984044775504083056, 7.89523115396791141333000559812, 9.18140853651376590241197577365, 10.7526131867532533766594939768, 11.23464566215086305632712958363, 11.77218865412510565581548253164, 12.75459866852370976899331361782, 13.23692020540698916819914096175, 14.13802890615593175052952149489, 15.16965670019614065584394731381, 15.88040672367580297108528371043, 16.300713053205811247923152301667, 17.20569558980037229330678147443, 18.06476141806277594804907636105, 18.86260031625512220017504745897, 19.638347815462901361785736386030, 20.62238003902459398156928083906, 21.4264457893091831171511243168

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