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Properties

Label 1-1183-1183.1130-r1-0-0
Degree $1$
Conductor $1183$
Sign $0.997 - 0.0714i$
Analytic cond. $127.131$
Root an. cond. $127.131$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.997 - 0.0714i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ 0.997 - 0.0714i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4969670899 + 0.01778298484i\)
\(L(\frac12)\) \(\approx\) \(0.4969670899 + 0.01778298484i\)
\(L(1)\) \(\approx\) \(0.4602086910 + 0.02876411717i\)
\(L(1)\) \(\approx\) \(0.4602086910 + 0.02876411717i\)

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

−21.02835867001120245278341504488, −20.143617500424707639613133251649, −19.48806162849284729720720139214, −18.31324900553841763141896740915, −17.958412471089366334463651644901, −17.0134308250677354090179529029, −16.55026610661459713559797189736, −15.72596416739593241813530707719, −15.461440586828249886886198882158, −14.13299325256654284526140713056, −13.1550070486346743584836101801, −12.38861369149706784777567841845, −11.31813028359957815337340464394, −10.76056573555899180152487881642, −9.95888899691351583579683047376, −8.963670444610217381436711123060, −8.50602565453342940747883167245, −7.3991864001805979927100084830, −6.48705234132872309590353266796, −5.74738735392821995862292437915, −4.90308399876623304982539205476, −4.35925856687220281526998497490, −2.57324757565141208031691257225, −1.25198020377554717413288978463, −0.41352763198433938495437913713, 0.370487772000288817261742677153, 1.88240679731192394761750997450, 2.36042409683937727680043911874, 3.69639116534306821610391893311, 4.5696895316511280166917426375, 5.961688682489095447429811734130, 6.60104562429279444388317024105, 7.59721068493331665918684268, 7.96613866624078146309566091270, 9.43773713630648932503165892196, 10.258889550518312244376673414543, 10.658596899799291097474089419890, 11.552455721738026456143298027615, 12.06016824950644615047238237126, 13.0886591873329220187281513597, 13.65191106009881386111647735740, 15.22806357630764929736453655903, 15.61159196980220849680893784252, 16.756743479843331578833494846196, 17.35182400465648994085031264446, 18.19243061229977975150740818398, 18.44615644908724629124230321565, 19.298469917423527524402748970497, 20.06201072259380869053540837046, 21.156975675789962799035412446711

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