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Properties

Label 1-1183-1183.1123-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: $\chi_{1183} (1123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.478 - 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2678763337 - 0.4511168273i\)
\(L(\frac12)\) \(\approx\) \(0.2678763337 - 0.4511168273i\)
\(L(1)\) \(\approx\) \(0.5803236800 - 0.1857734114i\)
\(L(1)\) \(\approx\) \(0.5803236800 - 0.1857734114i\)

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

−21.70528498452981348141443061905, −20.93509452193641649450994573287, −19.67741629869527163268334566495, −18.86115280100009900064531897480, −18.178730797153093312215873839884, −17.31535921440351281578701296425, −17.05562023507070245714328915885, −16.14337117398143300048660272570, −15.69019118801736246922100029591, −14.574132436489925114214583589842, −13.47357090175101776743347833624, −13.20094364825379393090308413076, −12.243268988280557615640248832539, −11.02251323829396365692734932598, −10.43853581809903797541075914460, −9.486906482143614487786973046333, −8.69289875369531997133681099737, −7.93558876523610945897939340447, −6.707529699610541045877561238098, −6.28834916240217482397225842510, −5.28324395256250684391255938827, −4.963674558211898576559299534103, −3.7532650266097162425830417293, −1.898253355481565083753358970966, −1.003495557361276488839977610224, 0.31675610503514680971144926094, 1.88890826960199130034488103497, 2.36704450633507266165525609809, 3.7336957808661618096236148885, 4.63540784474615371889392224453, 5.36091959658644904216692553544, 6.60382534850958618224813260805, 7.11184655848603130615473134802, 8.44464788625975579545897145554, 9.47651238465820514738481182864, 10.08860790557471787122875119125, 10.750226313073329449370095161814, 11.32777244456036916552899651772, 12.24004429264711505364947100217, 13.054103919883506669853856404538, 13.56441325154721447379559143294, 14.86121254144742920935488152826, 15.48851567521246151440825116051, 16.82701320978064921790839165404, 17.3392751160017430753131352441, 17.862655702239466287881617738812, 18.63946125629511587014972516774, 19.18287739627177991599695621360, 20.54012752095720421905404610996, 20.84612725953147129572993066863

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