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Properties

Label 1-1183-1183.1112-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.778 + 0.627i$
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  + (−0.721 + 0.692i)2-s + (−0.799 + 0.600i)3-s + (0.0402 − 0.999i)4-s + (0.822 − 0.568i)5-s + (0.160 − 0.987i)6-s + (0.663 + 0.748i)8-s + (0.278 − 0.960i)9-s + (−0.200 + 0.979i)10-s + (0.960 − 0.278i)11-s + (0.568 + 0.822i)12-s + (−0.316 + 0.948i)15-s + (−0.996 − 0.0804i)16-s + (−0.200 − 0.979i)17-s + (0.464 + 0.885i)18-s + (0.866 + 0.5i)19-s + (−0.534 − 0.845i)20-s + ⋯
L(s)  = 1  + (−0.721 + 0.692i)2-s + (−0.799 + 0.600i)3-s + (0.0402 − 0.999i)4-s + (0.822 − 0.568i)5-s + (0.160 − 0.987i)6-s + (0.663 + 0.748i)8-s + (0.278 − 0.960i)9-s + (−0.200 + 0.979i)10-s + (0.960 − 0.278i)11-s + (0.568 + 0.822i)12-s + (−0.316 + 0.948i)15-s + (−0.996 − 0.0804i)16-s + (−0.200 − 0.979i)17-s + (0.464 + 0.885i)18-s + (0.866 + 0.5i)19-s + (−0.534 − 0.845i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9753604435 + 0.3442034165i\)
\(L(\frac12)\) \(\approx\) \(0.9753604435 + 0.3442034165i\)
\(L(1)\) \(\approx\) \(0.7394359579 + 0.2239392578i\)
\(L(1)\) \(\approx\) \(0.7394359579 + 0.2239392578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.721 + 0.692i)T \)
3 \( 1 + (-0.799 + 0.600i)T \)
5 \( 1 + (0.822 - 0.568i)T \)
11 \( 1 + (0.960 - 0.278i)T \)
17 \( 1 + (-0.200 - 0.979i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.692 + 0.721i)T \)
31 \( 1 + (-0.935 + 0.354i)T \)
37 \( 1 + (0.774 + 0.632i)T \)
41 \( 1 + (-0.600 - 0.799i)T \)
43 \( 1 + (0.632 + 0.774i)T \)
47 \( 1 + (-0.464 + 0.885i)T \)
53 \( 1 + (-0.748 + 0.663i)T \)
59 \( 1 + (-0.0804 - 0.996i)T \)
61 \( 1 + (-0.948 + 0.316i)T \)
67 \( 1 + (0.999 - 0.0402i)T \)
71 \( 1 + (0.391 - 0.919i)T \)
73 \( 1 + (0.239 - 0.970i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (-0.992 - 0.120i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.903 - 0.428i)T \)
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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

−21.29843506371109190332394371676, −20.13523011800968810110029996712, −19.51752441801474732127582056366, −18.62010084880907265282041017223, −18.17447002198303848341882462546, −17.28344086134385955038751295095, −17.06751345673144734620976476734, −16.10228572894958893918056608761, −14.87903675149796176019444790977, −13.88027897216099829809444937346, −13.098682401552541721571275914682, −12.44664516265293657411719995852, −11.5302315082495040548718216423, −10.97906382746120276976475269177, −10.170879366247105896716187871399, −9.461651850069057823575241228913, −8.509484248897486307464847841987, −7.43326786206371541286915309122, −6.73014170743645491523897765439, −6.07545514736517187095808227110, −4.86806341911792137543254431720, −3.76054875727031808684284046927, −2.53979641394450084740178123768, −1.805908485761504264603107883229, −0.88969057410862897858386604594, 0.873874201566111761782968369672, 1.59345173872853553662784885989, 3.254857969603200104212552049668, 4.65404927974197036153470826754, 5.20080268573705511742470877636, 6.03558499621928973899950081426, 6.66306903562159007765213429353, 7.66377365311966439495705468197, 9.00020785863198544396806339454, 9.30872616549860272817541668192, 9.99579695790463687936147541644, 10.96097232555550959012039958678, 11.65604276430758985917326978117, 12.63651545377640055284113581621, 13.867767356951452686474310654, 14.34290179533914816627700846329, 15.43258316928726524513199055369, 16.20170511311657389510611060707, 16.65018813354111911903222449790, 17.384874230717590516189575773343, 17.95102343186028592414930828738, 18.65140598502150093535502962640, 19.86209290096247409138593976282, 20.43545141019819659759257287055, 21.34957555871380363127459385072

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