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Properties

Label 1-1183-1183.901-r1-0-0
Degree $1$
Conductor $1183$
Sign $0.960 - 0.277i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.692 + 0.721i)2-s + (−0.120 + 0.992i)3-s + (−0.0402 − 0.999i)4-s + (−0.996 − 0.0804i)5-s + (−0.632 − 0.774i)6-s + (0.748 + 0.663i)8-s + (−0.970 − 0.239i)9-s + (0.748 − 0.663i)10-s + (0.970 − 0.239i)11-s + (0.996 + 0.0804i)12-s + (0.200 − 0.979i)15-s + (−0.996 + 0.0804i)16-s + (0.200 − 0.979i)17-s + (0.845 − 0.534i)18-s + 19-s + (−0.0402 + 0.999i)20-s + ⋯
L(s)  = 1  + (−0.692 + 0.721i)2-s + (−0.120 + 0.992i)3-s + (−0.0402 − 0.999i)4-s + (−0.996 − 0.0804i)5-s + (−0.632 − 0.774i)6-s + (0.748 + 0.663i)8-s + (−0.970 − 0.239i)9-s + (0.748 − 0.663i)10-s + (0.970 − 0.239i)11-s + (0.996 + 0.0804i)12-s + (0.200 − 0.979i)15-s + (−0.996 + 0.0804i)16-s + (0.200 − 0.979i)17-s + (0.845 − 0.534i)18-s + 19-s + (−0.0402 + 0.999i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.960 - 0.277i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ 0.960 - 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6137770169 - 0.08701463453i\)
\(L(\frac12)\) \(\approx\) \(0.6137770169 - 0.08701463453i\)
\(L(1)\) \(\approx\) \(0.5348557278 + 0.2767191412i\)
\(L(1)\) \(\approx\) \(0.5348557278 + 0.2767191412i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.692 + 0.721i)T \)
3 \( 1 + (-0.120 + 0.992i)T \)
5 \( 1 + (-0.996 - 0.0804i)T \)
11 \( 1 + (0.970 - 0.239i)T \)
17 \( 1 + (0.200 - 0.979i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.278 + 0.960i)T \)
31 \( 1 + (-0.632 - 0.774i)T \)
37 \( 1 + (0.632 + 0.774i)T \)
41 \( 1 + (-0.919 + 0.391i)T \)
43 \( 1 + (0.987 + 0.160i)T \)
47 \( 1 + (-0.845 - 0.534i)T \)
53 \( 1 + (0.948 - 0.316i)T \)
59 \( 1 + (-0.996 - 0.0804i)T \)
61 \( 1 + (0.748 - 0.663i)T \)
67 \( 1 + (-0.885 + 0.464i)T \)
71 \( 1 + (-0.799 - 0.600i)T \)
73 \( 1 + (0.278 - 0.960i)T \)
79 \( 1 + (-0.845 - 0.534i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.996 + 0.0804i)T \)
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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

−20.72841001771431442648744920766, −19.85181194279994977539175308388, −19.64160792235090132829581501534, −18.82664375797869170952350623656, −18.185149300280262963080411054790, −17.40416600721596915895880337599, −16.68568268136690028624716988140, −15.927959572487164272428366583670, −14.72688582799873916520742303897, −13.96289920422005562126029378549, −12.86270847130854167680639215922, −12.26370860108868738225584457591, −11.734704212328049923832923351195, −11.03914970542920107379294965959, −10.11663842597305847041248682909, −8.9541261823686817841670017574, −8.350963140047777012573186018225, −7.53426563740447393421070240642, −6.962168150374616918383401562670, −5.94435671755019855688632650911, −4.43587246220765747299437528312, −3.63534799453012090535552071486, −2.69198272538316064145135009766, −1.60407426797896395239582987466, −0.79941218435519438181320477261, 0.236747266261356000304640290355, 1.27799157609545092065734635713, 3.02130909695298716411797554299, 3.90764440975121065081702442752, 4.83326633915564483578285067564, 5.5541212672074616970663327021, 6.61310912691449900775961417311, 7.49681863520930621947492373879, 8.281410306765780296239293349399, 9.19789951981286227859979976930, 9.61958493835271617813307661930, 10.6546268996062106328778067659, 11.56250810757962063232131869822, 11.83840444271504945760299602082, 13.550092186245462076374603577402, 14.405332156616802890488682038792, 14.986560007918911735541532969014, 15.78703249076305438720464285565, 16.35371916134232068301826620809, 16.80939634238342388983275376704, 17.85179564969814078734545961586, 18.56863667125472950936563715016, 19.63547627216989721476036000878, 20.00895333693547465813834969612, 20.72749209792511488607894249500

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