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Properties

Label 1-1183-1183.744-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.999 - 0.0388i$
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  + (−0.748 − 0.663i)2-s + (0.987 − 0.160i)3-s + (0.120 + 0.992i)4-s + (0.278 + 0.960i)5-s + (−0.845 − 0.534i)6-s + (0.568 − 0.822i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (0.948 + 0.316i)11-s + (0.278 + 0.960i)12-s + (0.428 + 0.903i)15-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (−0.919 − 0.391i)18-s + (−0.5 + 0.866i)19-s + (−0.919 + 0.391i)20-s + ⋯
L(s)  = 1  + (−0.748 − 0.663i)2-s + (0.987 − 0.160i)3-s + (0.120 + 0.992i)4-s + (0.278 + 0.960i)5-s + (−0.845 − 0.534i)6-s + (0.568 − 0.822i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (0.948 + 0.316i)11-s + (0.278 + 0.960i)12-s + (0.428 + 0.903i)15-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (−0.919 − 0.391i)18-s + (−0.5 + 0.866i)19-s + (−0.919 + 0.391i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.999 - 0.0388i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (744, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.999 - 0.0388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.830332241 - 0.03557330615i\)
\(L(\frac12)\) \(\approx\) \(1.830332241 - 0.03557330615i\)
\(L(1)\) \(\approx\) \(1.222924132 - 0.1239773133i\)
\(L(1)\) \(\approx\) \(1.222924132 - 0.1239773133i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.748 - 0.663i)T \)
3 \( 1 + (0.987 - 0.160i)T \)
5 \( 1 + (0.278 + 0.960i)T \)
11 \( 1 + (0.948 + 0.316i)T \)
17 \( 1 + (0.568 - 0.822i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.948 - 0.316i)T \)
31 \( 1 + (-0.0402 + 0.999i)T \)
37 \( 1 + (0.885 - 0.464i)T \)
41 \( 1 + (-0.632 - 0.774i)T \)
43 \( 1 + (-0.0402 - 0.999i)T \)
47 \( 1 + (-0.919 + 0.391i)T \)
53 \( 1 + (0.428 + 0.903i)T \)
59 \( 1 + (-0.970 - 0.239i)T \)
61 \( 1 + (-0.996 + 0.0804i)T \)
67 \( 1 + (-0.919 + 0.391i)T \)
71 \( 1 + (0.987 - 0.160i)T \)
73 \( 1 + (-0.200 - 0.979i)T \)
79 \( 1 + (-0.919 + 0.391i)T \)
83 \( 1 + (-0.354 - 0.935i)T \)
89 \( 1 + T \)
97 \( 1 + (0.692 + 0.721i)T \)
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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

−21.21187006721603548656211160551, −20.03558027025884605202625943457, −19.77774029665468913256649346638, −19.03839557107751624110740137122, −18.15859382583565961435616204918, −17.09342300770563214106167479416, −16.74759722899419370893617983018, −15.88273948677765907455484854959, −14.987298877106775422343149902453, −14.5402616472307940932971324347, −13.50770979660108820620982458728, −12.983283700172443470314971879726, −11.748275140976732494882204973889, −10.700941501732722736334975712589, −9.70466485060658961161952054104, −9.25900759745473805651714914770, −8.448205688822773660027184979577, −8.028832794939272395736084952639, −6.852386652966259680934963033892, −6.10979659534009193908660180712, −4.92775888278914483591887139778, −4.26314843062218921787415037560, −2.941769276659393432679622540572, −1.71362642238866220885544192698, −1.02218085964864408446265998661, 1.20927109012141783849414773784, 2.04995693555808908445187475324, 2.97359459425559125111126201120, 3.53664917542969474332226536560, 4.559062330506333087308231108, 6.28942740410610850338317754061, 7.087423338761966205492515153788, 7.65563351143908146140069536282, 8.703281792674470051211346569169, 9.34714545480645820780980691106, 10.101641195891740623527360087729, 10.73179049554208538601324228600, 11.854222922957362360486319565820, 12.4355762518815300167358348941, 13.51838198287192267960446420368, 14.1772175323583959300810418397, 14.8819359087565567566450030763, 15.79384165528065101681773593481, 16.81043925564776181130721967227, 17.61387061753909874864719863093, 18.44174267598220741124574455800, 18.914900080624074774992300244584, 19.604112989166564157032334076978, 20.27633094422370562098769387956, 21.19233205775647571054279516469

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