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Properties

Label 1-1183-1183.597-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.323 - 0.946i$
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  + (−0.428 + 0.903i)2-s + (−0.845 + 0.534i)3-s + (−0.632 − 0.774i)4-s + (−0.948 − 0.316i)5-s + (−0.120 − 0.992i)6-s + (0.970 − 0.239i)8-s + (0.428 − 0.903i)9-s + (0.692 − 0.721i)10-s + (−0.428 − 0.903i)11-s + (0.948 + 0.316i)12-s + (0.970 − 0.239i)15-s + (−0.200 + 0.979i)16-s + (0.278 − 0.960i)17-s + (0.632 + 0.774i)18-s + (0.5 + 0.866i)19-s + (0.354 + 0.935i)20-s + ⋯
L(s)  = 1  + (−0.428 + 0.903i)2-s + (−0.845 + 0.534i)3-s + (−0.632 − 0.774i)4-s + (−0.948 − 0.316i)5-s + (−0.120 − 0.992i)6-s + (0.970 − 0.239i)8-s + (0.428 − 0.903i)9-s + (0.692 − 0.721i)10-s + (−0.428 − 0.903i)11-s + (0.948 + 0.316i)12-s + (0.970 − 0.239i)15-s + (−0.200 + 0.979i)16-s + (0.278 − 0.960i)17-s + (0.632 + 0.774i)18-s + (0.5 + 0.866i)19-s + (0.354 + 0.935i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.323 - 0.946i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (597, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.323 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2570031621 - 0.1836659684i\)
\(L(\frac12)\) \(\approx\) \(0.2570031621 - 0.1836659684i\)
\(L(1)\) \(\approx\) \(0.4630031907 + 0.1326195604i\)
\(L(1)\) \(\approx\) \(0.4630031907 + 0.1326195604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.428 + 0.903i)T \)
3 \( 1 + (-0.845 + 0.534i)T \)
5 \( 1 + (-0.948 - 0.316i)T \)
11 \( 1 + (-0.428 - 0.903i)T \)
17 \( 1 + (0.278 - 0.960i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.568 + 0.822i)T \)
31 \( 1 + (0.919 + 0.391i)T \)
37 \( 1 + (-0.799 + 0.600i)T \)
41 \( 1 + (-0.885 - 0.464i)T \)
43 \( 1 + (0.120 - 0.992i)T \)
47 \( 1 + (0.632 - 0.774i)T \)
53 \( 1 + (0.278 - 0.960i)T \)
59 \( 1 + (0.200 + 0.979i)T \)
61 \( 1 + (0.278 + 0.960i)T \)
67 \( 1 + (-0.987 - 0.160i)T \)
71 \( 1 + (-0.885 - 0.464i)T \)
73 \( 1 + (-0.428 - 0.903i)T \)
79 \( 1 + (-0.632 + 0.774i)T \)
83 \( 1 + (-0.885 + 0.464i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.748 + 0.663i)T \)
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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

−21.4574233550962257637667000451, −20.41472144482012829273355685529, −19.635788245985827255495542917358, −19.1154473685836411116777434728, −18.40291923404180898677603305381, −17.549782768937208403056032435454, −17.22559935012353627239911799034, −15.98535828989625100565450418090, −15.468688701392384024206464037806, −14.171527188961503676102884495300, −13.19041518094095109013377401934, −12.560623048857139498921947460842, −11.787872307587058110220190011058, −11.37607374163472267634921985042, −10.41536998819517276517391838107, −9.88836652665286252027056252140, −8.56943447044898480736490889822, −7.6883985557370883174894053953, −7.294291581843637085895180694791, −6.116752656686236667144061394768, −4.84889187628874310965758523929, −4.24967985920533091924459014693, −3.10331399932834093562894395033, −2.1023560101902048727623137688, −1.04677807568200516700387019484, 0.23148539043697367153108460005, 1.16474285455885033883561839922, 3.21756127508462902028739384393, 4.15992518933524393198743645573, 5.0544409230188360486364692210, 5.59007377143692653581869334735, 6.64996969634484245701113281053, 7.375943169305289361242853365410, 8.40600764217021639624435142845, 8.90492055083656207318939096880, 10.19422715297443495501602180236, 10.51765156489929798488796054891, 11.72842143975408916720543476867, 12.18500171304282112017897393740, 13.45399384143866990977780396041, 14.29386994841678495429488037160, 15.2397154774356083901053023545, 15.92521114810901997562835095067, 16.36509698629576264278245324315, 16.88104376476749600809999022103, 17.99998141065988290678244949173, 18.54656664601967269223070111871, 19.26148823141534778684808436445, 20.34661480930784369492787579796, 21.00353815400889081053090352048

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