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Properties

Label 1-1183-1183.95-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.673 - 0.739i$
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  + (0.354 − 0.935i)2-s + (−0.996 + 0.0804i)3-s + (−0.748 − 0.663i)4-s + (−0.799 − 0.600i)5-s + (−0.278 + 0.960i)6-s + (−0.885 + 0.464i)8-s + (0.987 − 0.160i)9-s + (−0.845 + 0.534i)10-s + (−0.987 − 0.160i)11-s + (0.799 + 0.600i)12-s + (0.845 + 0.534i)15-s + (0.120 + 0.992i)16-s + (0.885 − 0.464i)17-s + (0.200 − 0.979i)18-s + (0.5 + 0.866i)19-s + (0.200 + 0.979i)20-s + ⋯
L(s)  = 1  + (0.354 − 0.935i)2-s + (−0.996 + 0.0804i)3-s + (−0.748 − 0.663i)4-s + (−0.799 − 0.600i)5-s + (−0.278 + 0.960i)6-s + (−0.885 + 0.464i)8-s + (0.987 − 0.160i)9-s + (−0.845 + 0.534i)10-s + (−0.987 − 0.160i)11-s + (0.799 + 0.600i)12-s + (0.845 + 0.534i)15-s + (0.120 + 0.992i)16-s + (0.885 − 0.464i)17-s + (0.200 − 0.979i)18-s + (0.5 + 0.866i)19-s + (0.200 + 0.979i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.673 - 0.739i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.673 - 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3389376875 - 0.7671933151i\)
\(L(\frac12)\) \(\approx\) \(0.3389376875 - 0.7671933151i\)
\(L(1)\) \(\approx\) \(0.5918291438 - 0.4494683786i\)
\(L(1)\) \(\approx\) \(0.5918291438 - 0.4494683786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.354 - 0.935i)T \)
3 \( 1 + (-0.996 + 0.0804i)T \)
5 \( 1 + (-0.799 - 0.600i)T \)
11 \( 1 + (-0.987 - 0.160i)T \)
17 \( 1 + (0.885 - 0.464i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.987 - 0.160i)T \)
31 \( 1 + (-0.692 - 0.721i)T \)
37 \( 1 + (0.970 - 0.239i)T \)
41 \( 1 + (-0.428 + 0.903i)T \)
43 \( 1 + (0.692 - 0.721i)T \)
47 \( 1 + (0.200 + 0.979i)T \)
53 \( 1 + (-0.845 - 0.534i)T \)
59 \( 1 + (-0.120 + 0.992i)T \)
61 \( 1 + (-0.0402 - 0.999i)T \)
67 \( 1 + (0.200 + 0.979i)T \)
71 \( 1 + (0.996 - 0.0804i)T \)
73 \( 1 + (0.632 - 0.774i)T \)
79 \( 1 + (-0.200 - 0.979i)T \)
83 \( 1 + (-0.568 + 0.822i)T \)
89 \( 1 - T \)
97 \( 1 + (0.919 + 0.391i)T \)
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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

−21.6802342691152488271434107347, −21.189956526241453195387912823968, −19.85871369903465152801174749897, −18.7474156728198281201963980021, −18.3736749272007236749042887952, −17.54977800454932480464241596222, −16.80159437342231657377299522218, −15.92230845834327163949320103341, −15.55616211960694282427436824953, −14.74087736911213622075699705285, −13.79893042338367269279023953941, −12.79418245856812736018454340358, −12.33161203339528805618679503270, −11.35100550359820413719445422226, −10.62353777577512946555059238210, −9.7024965761979503597141376003, −8.46202776402148875237608899111, −7.57600974670790521961695414404, −7.08646219104220217014273770681, −6.26367863598138659729904193431, −5.27169545105071774103541165481, −4.73754590852880718865240779303, −3.666467867653825060030633870245, −2.75518858464302156402124995450, −0.81635505189813695369696519106, 0.550551293341690944237104891769, 1.37548079577481394174361134118, 2.83904811829433428119978074044, 3.76721435537743478957153329422, 4.68982300230237298248243502799, 5.27241278223287783115502233837, 6.02214089959434310210272802715, 7.40570017681170501206723578829, 8.17019189471552342261702864826, 9.37414862259702303703787255628, 10.05659528224861388685412327835, 10.99072337384046891318791704300, 11.47619026909727960425696205559, 12.38313895196035199284947848084, 12.72022870134450467922776809113, 13.63899868555207500323323657263, 14.74445008022896301143712705568, 15.61391029689212226981952274209, 16.31181870023798386125497868378, 17.07275475750544018503683583782, 18.139426136812901325642158949458, 18.68658386704999297863632487317, 19.33653171217779898668017599896, 20.47432859867193707601179179090, 20.88127253674062181392843124849

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