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Properties

Label 1-1183-1183.1025-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.733 - 0.679i$
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  + (0.999 − 0.0402i)2-s + (0.970 + 0.239i)3-s + (0.996 − 0.0804i)4-s + (0.160 − 0.987i)5-s + (0.979 + 0.200i)6-s + (0.992 − 0.120i)8-s + (0.885 + 0.464i)9-s + (0.120 − 0.992i)10-s + (−0.464 − 0.885i)11-s + (0.987 + 0.160i)12-s + (0.391 − 0.919i)15-s + (0.987 − 0.160i)16-s + (−0.919 − 0.391i)17-s + (0.903 + 0.428i)18-s − i·19-s + (0.0804 − 0.996i)20-s + ⋯
L(s)  = 1  + (0.999 − 0.0402i)2-s + (0.970 + 0.239i)3-s + (0.996 − 0.0804i)4-s + (0.160 − 0.987i)5-s + (0.979 + 0.200i)6-s + (0.992 − 0.120i)8-s + (0.885 + 0.464i)9-s + (0.120 − 0.992i)10-s + (−0.464 − 0.885i)11-s + (0.987 + 0.160i)12-s + (0.391 − 0.919i)15-s + (0.987 − 0.160i)16-s + (−0.919 − 0.391i)17-s + (0.903 + 0.428i)18-s − i·19-s + (0.0804 − 0.996i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.733 - 0.679i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.733 - 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.074116040 - 1.596501824i\)
\(L(\frac12)\) \(\approx\) \(4.074116040 - 1.596501824i\)
\(L(1)\) \(\approx\) \(2.647679967 - 0.5108016919i\)
\(L(1)\) \(\approx\) \(2.647679967 - 0.5108016919i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.999 - 0.0402i)T \)
3 \( 1 + (0.970 + 0.239i)T \)
5 \( 1 + (0.160 - 0.987i)T \)
11 \( 1 + (-0.464 - 0.885i)T \)
17 \( 1 + (-0.919 - 0.391i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.845 + 0.534i)T \)
31 \( 1 + (0.979 + 0.200i)T \)
37 \( 1 + (0.979 + 0.200i)T \)
41 \( 1 + (-0.721 - 0.692i)T \)
43 \( 1 + (-0.948 - 0.316i)T \)
47 \( 1 + (-0.903 + 0.428i)T \)
53 \( 1 + (0.799 - 0.600i)T \)
59 \( 1 + (-0.160 + 0.987i)T \)
61 \( 1 + (-0.120 + 0.992i)T \)
67 \( 1 + (0.822 + 0.568i)T \)
71 \( 1 + (-0.960 + 0.278i)T \)
73 \( 1 + (0.534 - 0.845i)T \)
79 \( 1 + (0.428 + 0.903i)T \)
83 \( 1 + (0.239 + 0.970i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.160 - 0.987i)T \)
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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

−21.39620267166650792613430589662, −20.50406023549758304592680080778, −20.07272123281124170338369925890, −19.05794358429182956003810001119, −18.48399836229548591831420203303, −17.54354060910443012397759487804, −16.45027467677339167128574381542, −15.33071876472258587253220735230, −15.02492534529832099907889631195, −14.42854962125113811789569584767, −13.50002062407107044998556085834, −13.02465045926793746385175570057, −12.15156920215982839889166963884, −11.17720161416453543245704281844, −10.280700442145303034827786488581, −9.674543171297928884722722570416, −8.23555900084793966751691846945, −7.62917074709359362595536625610, −6.71274445824357250002435925535, −6.23740640747913995746633574316, −4.859083444153870094043060921901, −4.02175867777775057903724323928, −3.17110834945187566040868307372, −2.34114295510311766670574149001, −1.76329555363806713085352342887, 1.14476782200796001820994042280, 2.235594856693933762070953646376, 3.037522654781404370902979337454, 3.94824681745285294595513626072, 4.83869810274022829045020840406, 5.3887553202453677832058896169, 6.608370471207748386185049309800, 7.52076353275998793046261358688, 8.43955498041481282038660889955, 9.10840564860821555849376377, 10.0723961181225972601583543277, 11.075705092084313867312911442806, 11.81596721023426165863296486666, 13.05523668395655555104589988931, 13.31488815065671748218258453986, 13.856525354695180285518448171717, 14.98518938692831781537061596926, 15.56825252343009410082702522726, 16.20847572524119988265232106038, 16.946797136978600403867756312120, 18.13270672217799050932918119442, 19.310007250537614429651011357611, 19.79449073588570486721712924365, 20.498826881955559743928470441954, 21.17806170736104813634227474758

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