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Properties

Label 1-1183-1183.1054-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.915 + 0.401i$
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  + (−0.845 + 0.534i)2-s + (0.278 + 0.960i)3-s + (0.428 − 0.903i)4-s + (0.987 + 0.160i)5-s + (−0.748 − 0.663i)6-s + (0.120 + 0.992i)8-s + (−0.845 + 0.534i)9-s + (−0.919 + 0.391i)10-s + (−0.845 − 0.534i)11-s + (0.987 + 0.160i)12-s + (0.120 + 0.992i)15-s + (−0.632 − 0.774i)16-s + (0.799 − 0.600i)17-s + (0.428 − 0.903i)18-s + (−0.5 + 0.866i)19-s + (0.568 − 0.822i)20-s + ⋯
L(s)  = 1  + (−0.845 + 0.534i)2-s + (0.278 + 0.960i)3-s + (0.428 − 0.903i)4-s + (0.987 + 0.160i)5-s + (−0.748 − 0.663i)6-s + (0.120 + 0.992i)8-s + (−0.845 + 0.534i)9-s + (−0.919 + 0.391i)10-s + (−0.845 − 0.534i)11-s + (0.987 + 0.160i)12-s + (0.120 + 0.992i)15-s + (−0.632 − 0.774i)16-s + (0.799 − 0.600i)17-s + (0.428 − 0.903i)18-s + (−0.5 + 0.866i)19-s + (0.568 − 0.822i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.915 + 0.401i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1054, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.915 + 0.401i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2097809338 + 1.000226368i\)
\(L(\frac12)\) \(\approx\) \(0.2097809338 + 1.000226368i\)
\(L(1)\) \(\approx\) \(0.6690341967 + 0.5163539980i\)
\(L(1)\) \(\approx\) \(0.6690341967 + 0.5163539980i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.845 + 0.534i)T \)
3 \( 1 + (0.278 + 0.960i)T \)
5 \( 1 + (0.987 + 0.160i)T \)
11 \( 1 + (-0.845 - 0.534i)T \)
17 \( 1 + (0.799 - 0.600i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.885 + 0.464i)T \)
31 \( 1 + (-0.200 + 0.979i)T \)
37 \( 1 + (0.948 - 0.316i)T \)
41 \( 1 + (-0.970 - 0.239i)T \)
43 \( 1 + (-0.748 + 0.663i)T \)
47 \( 1 + (0.428 + 0.903i)T \)
53 \( 1 + (0.799 - 0.600i)T \)
59 \( 1 + (-0.632 + 0.774i)T \)
61 \( 1 + (0.799 + 0.600i)T \)
67 \( 1 + (-0.996 - 0.0804i)T \)
71 \( 1 + (-0.970 - 0.239i)T \)
73 \( 1 + (-0.845 - 0.534i)T \)
79 \( 1 + (0.428 + 0.903i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.354 + 0.935i)T \)
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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

−20.575677420071574205837671348810, −20.19134608316538550429194162854, −19.21539512613038758981517607186, −18.43511367646128300980575288892, −18.09311739929080125795733854959, −17.144402340427861584482859608436, −16.81566358381324320896747121396, −15.503477676681828905086219767531, −14.606310513721249423531598727452, −13.49178788976952137639350771969, −13.03969863303571602540908987684, −12.33924565592248451959979832218, −11.523183633826307703989822502633, −10.35597289414507687696511245665, −9.94704240529379653951584159543, −8.83939869555555967146703397201, −8.28632330912656066736289648367, −7.41594547065909032738628156922, −6.57220384121969988044117761897, −5.78861355335196560765273015641, −4.47784446937163524827279139030, −3.0481308973343048157157005738, −2.32979881301040849917231108887, −1.6902951318260714909457250472, −0.51856719309225535786747484037, 1.33205896821272761561864455172, 2.47738195885163280756260956470, 3.27223289920633353008958707207, 4.78279093302460820965375051422, 5.55069422789985512084896576377, 6.07525720679985447998020140550, 7.31060032840702584944979110151, 8.220667604364121427533192515755, 8.88964874521865246994541229705, 9.80283893497222984661186376725, 10.24005545434400953154176695650, 10.86286439760255288928599125822, 11.9092681288844624232719376924, 13.34123016739684871190960420535, 14.11092568767132677487217559663, 14.62486537541489746769458740942, 15.53346808879690374323445578249, 16.37677741818368830219040986385, 16.66781690013409001355864089403, 17.762138254885426792428698832664, 18.27370690437214938856536742160, 19.188244742019828978750462826298, 19.98169577561116303055419144158, 20.962520360372324097076535147710, 21.2377906771311229806242631492

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