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Properties

Label 1-1183-1183.51-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.707 + 0.707i$
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  + (−0.799 + 0.600i)2-s + (0.948 − 0.316i)3-s + (0.278 − 0.960i)4-s + (0.0402 + 0.999i)5-s + (−0.568 + 0.822i)6-s + (0.354 + 0.935i)8-s + (0.799 − 0.600i)9-s + (−0.632 − 0.774i)10-s + (−0.799 − 0.600i)11-s + (−0.0402 − 0.999i)12-s + (0.354 + 0.935i)15-s + (−0.845 − 0.534i)16-s + (0.987 + 0.160i)17-s + (−0.278 + 0.960i)18-s + (0.5 + 0.866i)19-s + (0.970 + 0.239i)20-s + ⋯
L(s)  = 1  + (−0.799 + 0.600i)2-s + (0.948 − 0.316i)3-s + (0.278 − 0.960i)4-s + (0.0402 + 0.999i)5-s + (−0.568 + 0.822i)6-s + (0.354 + 0.935i)8-s + (0.799 − 0.600i)9-s + (−0.632 − 0.774i)10-s + (−0.799 − 0.600i)11-s + (−0.0402 − 0.999i)12-s + (0.354 + 0.935i)15-s + (−0.845 − 0.534i)16-s + (0.987 + 0.160i)17-s + (−0.278 + 0.960i)18-s + (0.5 + 0.866i)19-s + (0.970 + 0.239i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.379504565 + 0.5712977764i\)
\(L(\frac12)\) \(\approx\) \(1.379504565 + 0.5712977764i\)
\(L(1)\) \(\approx\) \(1.022991795 + 0.2692514585i\)
\(L(1)\) \(\approx\) \(1.022991795 + 0.2692514585i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.799 + 0.600i)T \)
3 \( 1 + (0.948 - 0.316i)T \)
5 \( 1 + (0.0402 + 0.999i)T \)
11 \( 1 + (-0.799 - 0.600i)T \)
17 \( 1 + (0.987 + 0.160i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.120 + 0.992i)T \)
31 \( 1 + (-0.428 - 0.903i)T \)
37 \( 1 + (0.996 + 0.0804i)T \)
41 \( 1 + (0.748 + 0.663i)T \)
43 \( 1 + (0.568 + 0.822i)T \)
47 \( 1 + (-0.278 - 0.960i)T \)
53 \( 1 + (0.987 + 0.160i)T \)
59 \( 1 + (0.845 - 0.534i)T \)
61 \( 1 + (0.987 - 0.160i)T \)
67 \( 1 + (-0.692 + 0.721i)T \)
71 \( 1 + (0.748 + 0.663i)T \)
73 \( 1 + (-0.799 - 0.600i)T \)
79 \( 1 + (0.278 + 0.960i)T \)
83 \( 1 + (0.748 - 0.663i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.885 + 0.464i)T \)
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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

−20.97983408835908649927332985176, −20.326271648554649701158323151673, −19.67573586224069965124376858113, −19.09114541957142452855200640579, −18.04653793313189929072369189693, −17.48352497370067908046063063132, −16.35539566429739663504879907004, −15.94353161420387069747741902646, −15.16209969560826234024479621989, −13.89659682309716838877320457951, −13.22509093353218895216886459289, −12.527722009023555014687128022404, −11.74718682206023640528566068791, −10.63081181370724461651217038070, −9.78075040581795843646472460754, −9.376060646848315866086538474015, −8.521088839697310752605857297077, −7.74372160811644913485992383237, −7.276388520422527148788751733168, −5.55687820090701169590459755019, −4.58038743712511874235000307672, −3.75216756692992200464756021093, −2.73467856438040264026654461834, −1.94424843488505781976671152968, −0.885754770138435137756144202416, 0.99872985235203405111304719830, 2.2004046547978610470097760546, 2.918364785861751990543747425341, 3.94590421985889061965110398247, 5.47563053824524795899357726518, 6.2280665118300455946330691495, 7.12145205624393814160317476762, 7.89397466513878583562596206821, 8.25950366955551160478453634563, 9.45838003819624952736213277044, 10.079446503311916641434020945882, 10.75789844566199856163347344529, 11.777757423823686484247495287210, 12.9636408959897158149508931534, 13.86872205070259827191928357421, 14.64371850435756560350441591466, 14.834148167511960078612631389528, 16.05783473375589967317736297260, 16.45422590468828318329091705888, 17.837051869058553696999819214143, 18.40627045084418362785841672895, 18.776828648192546493173860386640, 19.520844075922050637680124299134, 20.37744823792381462440403872869, 21.087140655161550278047349940592

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