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Properties

Label 1-1183-1183.1074-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.478 - 0.878i$
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  + (0.316 + 0.948i)2-s + (−0.987 + 0.160i)3-s + (−0.799 + 0.600i)4-s + (−0.721 − 0.692i)5-s + (−0.464 − 0.885i)6-s + (−0.822 − 0.568i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (0.316 − 0.948i)11-s + (0.692 − 0.721i)12-s + (0.822 + 0.568i)15-s + (0.278 − 0.960i)16-s + (−0.996 − 0.0804i)17-s + (0.600 + 0.799i)18-s + (0.866 + 0.5i)19-s + (0.992 + 0.120i)20-s + ⋯
L(s)  = 1  + (0.316 + 0.948i)2-s + (−0.987 + 0.160i)3-s + (−0.799 + 0.600i)4-s + (−0.721 − 0.692i)5-s + (−0.464 − 0.885i)6-s + (−0.822 − 0.568i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (0.316 − 0.948i)11-s + (0.692 − 0.721i)12-s + (0.822 + 0.568i)15-s + (0.278 − 0.960i)16-s + (−0.996 − 0.0804i)17-s + (0.600 + 0.799i)18-s + (0.866 + 0.5i)19-s + (0.992 + 0.120i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.478 - 0.878i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1074, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.478 - 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06175924013 - 0.1040055762i\)
\(L(\frac12)\) \(\approx\) \(0.06175924013 - 0.1040055762i\)
\(L(1)\) \(\approx\) \(0.5822853725 + 0.2022761518i\)
\(L(1)\) \(\approx\) \(0.5822853725 + 0.2022761518i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.316 + 0.948i)T \)
3 \( 1 + (-0.987 + 0.160i)T \)
5 \( 1 + (-0.721 - 0.692i)T \)
11 \( 1 + (0.316 - 0.948i)T \)
17 \( 1 + (-0.996 - 0.0804i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (-0.534 + 0.845i)T \)
37 \( 1 + (0.999 + 0.0402i)T \)
41 \( 1 + (0.935 + 0.354i)T \)
43 \( 1 + (-0.885 - 0.464i)T \)
47 \( 1 + (-0.600 + 0.799i)T \)
53 \( 1 + (-0.996 - 0.0804i)T \)
59 \( 1 + (-0.960 + 0.278i)T \)
61 \( 1 + (0.996 - 0.0804i)T \)
67 \( 1 + (-0.391 - 0.919i)T \)
71 \( 1 + (-0.935 - 0.354i)T \)
73 \( 1 + (-0.316 + 0.948i)T \)
79 \( 1 + (0.799 + 0.600i)T \)
83 \( 1 + (-0.935 + 0.354i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.239 + 0.970i)T \)
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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

−21.85362941122904941597930474156, −20.631812131898681846538632262460, −19.88566271062380312737021853590, −19.276718275531344956427257801389, −18.29145662732056235671728003187, −17.969371040459642091322437748258, −17.117794682513688963581878634158, −15.94751109053931355796357859965, −15.18597104333977271808111262512, −14.54044934014946185289435732884, −13.30959517679546242414452017365, −12.81964742771483452700107817305, −11.781823795950174620880686762190, −11.406455908666298051878434145442, −10.792163575493178835609877758164, −9.849978985494843103108911514898, −9.146270914715068755693172721910, −7.68945695724679695257058056096, −6.99050090525056650669589355645, −6.064369631430240685424384551439, −5.024315383208052479479282341446, −4.33256062066219255077563507793, −3.48207702573255575225719614914, −2.33215097894561195304751049076, −1.33219689961914489140258958276, 0.06052221465534319267215417626, 1.17309646360369881826109253997, 3.21208802277894632696982169938, 4.13712099689121314316342650997, 4.76467534048611347916373978340, 5.61081169508947020389259484426, 6.34209482126585560068626848629, 7.21737725504950593234426911619, 8.06858070596832610920119457793, 8.92568791382269851535485279467, 9.664448553661831075560222822720, 11.05038961063475138739647562580, 11.572035029025321263428605161867, 12.5321857032542847864273388375, 13.04021133631810206936790604192, 14.01957341457815398596422257195, 15.034786483475483091777489708080, 15.75169549473378669279929747358, 16.46276532626785730195729194261, 16.71183599466829575861190150208, 17.69898736025556336433148089065, 18.42890923334223974620006802735, 19.22399154166068284073731450623, 20.36545205765540546586815433239, 21.20195100818999214549351625468

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