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Properties

Label 1-1183-1183.241-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.695 + 0.718i$
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  + (0.935 − 0.354i)2-s + (−0.428 − 0.903i)3-s + (0.748 − 0.663i)4-s + (−0.391 + 0.919i)5-s + (−0.721 − 0.692i)6-s + (0.464 − 0.885i)8-s + (−0.632 + 0.774i)9-s + (−0.0402 + 0.999i)10-s + (−0.774 + 0.632i)11-s + (−0.919 − 0.391i)12-s + (0.999 − 0.0402i)15-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + (−0.316 + 0.948i)18-s + (−0.866 + 0.5i)19-s + (0.316 + 0.948i)20-s + ⋯
L(s)  = 1  + (0.935 − 0.354i)2-s + (−0.428 − 0.903i)3-s + (0.748 − 0.663i)4-s + (−0.391 + 0.919i)5-s + (−0.721 − 0.692i)6-s + (0.464 − 0.885i)8-s + (−0.632 + 0.774i)9-s + (−0.0402 + 0.999i)10-s + (−0.774 + 0.632i)11-s + (−0.919 − 0.391i)12-s + (0.999 − 0.0402i)15-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + (−0.316 + 0.948i)18-s + (−0.866 + 0.5i)19-s + (0.316 + 0.948i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.695 + 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.269934328 + 0.5385384357i\)
\(L(\frac12)\) \(\approx\) \(1.269934328 + 0.5385384357i\)
\(L(1)\) \(\approx\) \(1.240880447 - 0.2118505077i\)
\(L(1)\) \(\approx\) \(1.240880447 - 0.2118505077i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.935 - 0.354i)T \)
3 \( 1 + (-0.428 - 0.903i)T \)
5 \( 1 + (-0.391 + 0.919i)T \)
11 \( 1 + (-0.774 + 0.632i)T \)
17 \( 1 + (0.885 + 0.464i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.632 + 0.774i)T \)
31 \( 1 + (0.960 - 0.278i)T \)
37 \( 1 + (-0.239 + 0.970i)T \)
41 \( 1 + (-0.0804 + 0.996i)T \)
43 \( 1 + (-0.278 + 0.960i)T \)
47 \( 1 + (0.316 + 0.948i)T \)
53 \( 1 + (-0.0402 - 0.999i)T \)
59 \( 1 + (-0.992 + 0.120i)T \)
61 \( 1 + (0.845 + 0.534i)T \)
67 \( 1 + (0.316 + 0.948i)T \)
71 \( 1 + (-0.903 + 0.428i)T \)
73 \( 1 + (0.160 + 0.987i)T \)
79 \( 1 + (0.948 - 0.316i)T \)
83 \( 1 + (0.822 - 0.568i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.600 - 0.799i)T \)
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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

−21.194016719297623632325551388402, −20.699008826872275922468139386957, −19.9744919107943717590605784728, −18.95446664126031509421007146892, −17.64554380032354575317621337103, −16.89149551096473016959666689860, −16.37218214264308363288831567392, −15.62103465492860735377817607276, −15.25655621729806063571137131957, −14.08056600295816706620360466369, −13.45009489273051816754557959438, −12.35352335345271789237483044756, −11.96368728242095690489548453128, −11.0207429551391750446696852925, −10.29238845435332639806478440831, −9.11414228230205095558874599581, −8.30663057389368730562605341870, −7.53574976472750445118018246619, −6.244472643494040504633364054, −5.508288691992427544324696687372, −4.92940052340994778022179900295, −4.060915406300754352551288429288, −3.39990327106822832891977463607, −2.20842910572805289950872752114, −0.40745117608910983557401164290, 1.38691536479406442933261771438, 2.3054884942708489891163638720, 3.06949665663069312563188490587, 4.1333743948104326299025216865, 5.14266419330398544050497675097, 6.10164112989408365398983103151, 6.60570527748645712488972959005, 7.61082262341270563344071894980, 8.0939593998645407190343558766, 10.00649386367667089501932616640, 10.4518154171512644205565700392, 11.359982772119253022025362886044, 12.000904403007350154889955375022, 12.71930634152494411079974436046, 13.36469894262019568317751321975, 14.38755145573674321560611635551, 14.78624915213485366834354301012, 15.76353054601810807351783595380, 16.59236336123736861993248898818, 17.65849001572284151201920604609, 18.52116947923845301876236364082, 18.995556998265274792904592687313, 19.72107568763075597565727761407, 20.58145015977125148058704051765, 21.46898058731186431020681952180

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