Properties

Label 1-1183-1183.478-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.805 + 0.592i$
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  + (−0.885 + 0.464i)2-s + (0.692 + 0.721i)3-s + (0.568 − 0.822i)4-s + (−0.987 + 0.160i)5-s + (−0.948 − 0.316i)6-s + (−0.120 + 0.992i)8-s + (−0.0402 + 0.999i)9-s + (0.799 − 0.600i)10-s + (0.0402 + 0.999i)11-s + (0.987 − 0.160i)12-s + (−0.799 − 0.600i)15-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + (−0.428 − 0.903i)18-s + (0.5 − 0.866i)19-s + (−0.428 + 0.903i)20-s + ⋯
L(s)  = 1  + (−0.885 + 0.464i)2-s + (0.692 + 0.721i)3-s + (0.568 − 0.822i)4-s + (−0.987 + 0.160i)5-s + (−0.948 − 0.316i)6-s + (−0.120 + 0.992i)8-s + (−0.0402 + 0.999i)9-s + (0.799 − 0.600i)10-s + (0.0402 + 0.999i)11-s + (0.987 − 0.160i)12-s + (−0.799 − 0.600i)15-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + (−0.428 − 0.903i)18-s + (0.5 − 0.866i)19-s + (−0.428 + 0.903i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.805 + 0.592i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (478, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.805 + 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2905750150 + 0.8861418628i\)
\(L(\frac12)\) \(\approx\) \(0.2905750150 + 0.8861418628i\)
\(L(1)\) \(\approx\) \(0.6425860597 + 0.4354404915i\)
\(L(1)\) \(\approx\) \(0.6425860597 + 0.4354404915i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.885 + 0.464i)T \)
3 \( 1 + (0.692 + 0.721i)T \)
5 \( 1 + (-0.987 + 0.160i)T \)
11 \( 1 + (0.0402 + 0.999i)T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.0402 + 0.999i)T \)
31 \( 1 + (0.200 + 0.979i)T \)
37 \( 1 + (0.748 - 0.663i)T \)
41 \( 1 + (-0.278 + 0.960i)T \)
43 \( 1 + (-0.200 + 0.979i)T \)
47 \( 1 + (-0.428 + 0.903i)T \)
53 \( 1 + (0.799 + 0.600i)T \)
59 \( 1 + (0.354 - 0.935i)T \)
61 \( 1 + (-0.919 - 0.391i)T \)
67 \( 1 + (-0.428 + 0.903i)T \)
71 \( 1 + (-0.692 - 0.721i)T \)
73 \( 1 + (0.845 - 0.534i)T \)
79 \( 1 + (0.428 - 0.903i)T \)
83 \( 1 + (0.970 + 0.239i)T \)
89 \( 1 - T \)
97 \( 1 + (0.632 - 0.774i)T \)
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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

−20.71369986551915356347670894002, −19.91602666134748493869746989915, −19.230661827156031728132757977663, −18.8700997943560997933452686401, −18.203929705606922106783010073096, −17.00337146196995834211186582852, −16.594825174236365826992527955923, −15.41617936571210158320594739003, −14.96036031993025833281174395309, −13.67254521610447067632768905130, −13.00486259262119334295830912322, −12.08914111297356969212161149615, −11.62616198889462764233199961859, −10.707186307409182616913407040922, −9.709080466363269423248685887301, −8.70493693933651898140357522644, −8.274025881467775246084173581212, −7.62180145015588089877084517820, −6.79488563432912898620794008655, −5.799400643586464717180324256955, −4.01810517281813935137542885180, −3.489477451277962212598795859332, −2.58924599529806295172995958033, −1.43941341085290053628324846888, −0.53884457295474480967113443355, 1.18172994345447682204993019144, 2.580123075385626648825364262685, 3.25197267920369254752666675421, 4.70212603561018011808309520847, 5.00273604418266001244200124721, 6.65564222370148275181724486009, 7.3928547906103037374970244481, 7.90887541009744719283960572040, 9.02200207131104798012744868785, 9.356820896074552779176893619122, 10.38264500523748636062254102716, 11.081871238073572085219502164068, 11.82691480719243665575843168098, 13.02235338712668888449532709309, 14.23583591139028033021469047835, 14.80172039339873857734375486955, 15.44526420366177965539346256266, 16.082325519845078512872822926226, 16.63951954385349517233204808867, 17.79824915056258881498695277737, 18.42338762624379128635901520375, 19.41549441284769922425441811482, 19.88503555423084348263597436964, 20.411572455272771477843869162484, 21.27288420190831170366304364695

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