↑

Properties

Label 1-1183-1183.545-r1-0-0
Degree $1$
Conductor $1183$
Sign $0.641 + 0.766i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.885 + 0.464i)2-s + (0.970 − 0.239i)3-s + (0.568 − 0.822i)4-s + (−0.354 + 0.935i)5-s + (−0.748 + 0.663i)6-s + (−0.120 + 0.992i)8-s + (0.885 − 0.464i)9-s + (−0.120 − 0.992i)10-s + (−0.885 − 0.464i)11-s + (0.354 − 0.935i)12-s + (−0.120 + 0.992i)15-s + (−0.354 − 0.935i)16-s + (−0.120 + 0.992i)17-s + (−0.568 + 0.822i)18-s + 19-s + (0.568 + 0.822i)20-s + ⋯
L(s)  = 1  + (−0.885 + 0.464i)2-s + (0.970 − 0.239i)3-s + (0.568 − 0.822i)4-s + (−0.354 + 0.935i)5-s + (−0.748 + 0.663i)6-s + (−0.120 + 0.992i)8-s + (0.885 − 0.464i)9-s + (−0.120 − 0.992i)10-s + (−0.885 − 0.464i)11-s + (0.354 − 0.935i)12-s + (−0.120 + 0.992i)15-s + (−0.354 − 0.935i)16-s + (−0.120 + 0.992i)17-s + (−0.568 + 0.822i)18-s + 19-s + (0.568 + 0.822i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.641 + 0.766i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ 0.641 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.690769384 + 0.7894818711i\)
\(L(\frac12)\) \(\approx\) \(1.690769384 + 0.7894818711i\)
\(L(1)\) \(\approx\) \(0.9582148285 + 0.2333392663i\)
\(L(1)\) \(\approx\) \(0.9582148285 + 0.2333392663i\)

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 \( 1 + (0.970 - 0.239i)T \)
5 \( 1 + (-0.354 + 0.935i)T \)
11 \( 1 + (-0.885 - 0.464i)T \)
17 \( 1 + (-0.120 + 0.992i)T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (0.885 - 0.464i)T \)
31 \( 1 + (-0.748 + 0.663i)T \)
37 \( 1 + (0.748 - 0.663i)T \)
41 \( 1 + (-0.970 + 0.239i)T \)
43 \( 1 + (-0.748 - 0.663i)T \)
47 \( 1 + (0.568 + 0.822i)T \)
53 \( 1 + (0.120 - 0.992i)T \)
59 \( 1 + (-0.354 + 0.935i)T \)
61 \( 1 + (-0.120 - 0.992i)T \)
67 \( 1 + (-0.568 - 0.822i)T \)
71 \( 1 + (0.970 - 0.239i)T \)
73 \( 1 + (0.885 + 0.464i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (-0.970 - 0.239i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.354 - 0.935i)T \)
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

−20.50751391528831994876371336341, −20.29915959595935031638945787159, −19.59175509452726356106546642907, −18.57815388239327055841192925625, −18.193310504822080487886556530201, −16.99926110762504554897910655712, −16.27869908911895200882403745308, −15.67144602020235895810263074200, −15.020196142554982560211879015265, −13.6320761409688463249576811214, −13.11312873336053858465349528553, −12.2706415910543420840439494191, −11.463489854761642105396670654958, −10.43530068393360171116513386897, −9.623454019248681862325574729991, −9.098756472355703777421608413158, −8.28577313700513213662329811309, −7.61872239836555801217683625196, −6.97555725065201345542193166335, −5.21398381944966711460120772423, −4.47848181258418278378388042504, −3.3447853607241701574329026152, −2.65466796703470367429298821174, −1.58374231223619320271647905912, −0.622058756372459009291153104033, 0.73830161960733528789787534509, 1.93995269846819730880308263834, 2.83223716076994131214403329507, 3.56019439179696137284044587652, 5.01411975764096694495474155171, 6.17111926700899444458688381161, 6.95216932968837533958905321217, 7.683144195082309288627611625837, 8.246495995863240092962841654938, 9.07719557442447848665932567343, 10.03859553754532620135169105979, 10.637000978941447030087832942804, 11.45373765804487691053889553060, 12.60110994597848130490419223453, 13.67850773180039171964328347249, 14.28979014159339752481479342879, 15.16729486884269662835995692861, 15.53025150146719583526374659329, 16.38880659539528635186914946006, 17.50478052090504470833921925257, 18.40977897170020051722801027278, 18.60890501258683566103238637769, 19.579637758409419445282828505153, 19.926118500343389684101972602270, 21.019508439949038004806109951585

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