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Properties

Label 1-1183-1183.101-r1-0-0
Degree $1$
Conductor $1183$
Sign $0.0245 + 0.999i$
Analytic cond. $127.131$
Root an. cond. $127.131$
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  + (0.200 − 0.979i)2-s + (0.354 − 0.935i)3-s + (−0.919 − 0.391i)4-s + (0.692 − 0.721i)5-s + (−0.845 − 0.534i)6-s + (−0.568 + 0.822i)8-s + (−0.748 − 0.663i)9-s + (−0.568 − 0.822i)10-s + (0.748 − 0.663i)11-s + (−0.692 + 0.721i)12-s + (−0.428 − 0.903i)15-s + (0.692 + 0.721i)16-s + (−0.428 − 0.903i)17-s + (−0.799 + 0.600i)18-s + 19-s + (−0.919 + 0.391i)20-s + ⋯
L(s)  = 1  + (0.200 − 0.979i)2-s + (0.354 − 0.935i)3-s + (−0.919 − 0.391i)4-s + (0.692 − 0.721i)5-s + (−0.845 − 0.534i)6-s + (−0.568 + 0.822i)8-s + (−0.748 − 0.663i)9-s + (−0.568 − 0.822i)10-s + (0.748 − 0.663i)11-s + (−0.692 + 0.721i)12-s + (−0.428 − 0.903i)15-s + (0.692 + 0.721i)16-s + (−0.428 − 0.903i)17-s + (−0.799 + 0.600i)18-s + 19-s + (−0.919 + 0.391i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.0245 + 0.999i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ 0.0245 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.657598432 - 1.698731002i\)
\(L(\frac12)\) \(\approx\) \(-1.657598432 - 1.698731002i\)
\(L(1)\) \(\approx\) \(0.4939629613 - 1.230143728i\)
\(L(1)\) \(\approx\) \(0.4939629613 - 1.230143728i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.200 - 0.979i)T \)
3 \( 1 + (0.354 - 0.935i)T \)
5 \( 1 + (0.692 - 0.721i)T \)
11 \( 1 + (0.748 - 0.663i)T \)
17 \( 1 + (-0.428 - 0.903i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.948 - 0.316i)T \)
31 \( 1 + (-0.845 - 0.534i)T \)
37 \( 1 + (0.845 + 0.534i)T \)
41 \( 1 + (-0.632 - 0.774i)T \)
43 \( 1 + (-0.0402 - 0.999i)T \)
47 \( 1 + (0.799 + 0.600i)T \)
53 \( 1 + (-0.996 - 0.0804i)T \)
59 \( 1 + (0.692 - 0.721i)T \)
61 \( 1 + (-0.568 - 0.822i)T \)
67 \( 1 + (-0.120 + 0.992i)T \)
71 \( 1 + (-0.987 + 0.160i)T \)
73 \( 1 + (0.948 + 0.316i)T \)
79 \( 1 + (0.799 + 0.600i)T \)
83 \( 1 + (-0.354 - 0.935i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−21.83729471073942403622381642364, −21.19914772191715331509015444419, −20.02538986382084355756185623803, −19.40020514208520185649235885110, −18.07678075304600313381244362082, −17.74246364403973016774730372283, −16.83149246557599224949346100474, −16.162199861850287595899794164078, −15.18598133780754654954403253637, −14.82415713533595250609055025392, −14.02720666029534981126023901142, −13.51083837534442922979478599610, −12.37845723808708095706064683720, −11.27989486737486429711465889280, −10.26847922710055788902591333968, −9.58882478787796594820200533590, −9.04299010729469229344180071517, −7.99785677453306906987839886069, −7.12835777482201607086974018879, −6.23345006420105358104677527060, −5.50020264664834061885644600648, −4.56743066392052641697229960924, −3.71913435013314489617068254682, −2.92517688813904663878200952052, −1.58599746092957995524265500103, 0.44915073671999705111225796185, 1.06981932828613159861462897998, 2.04654586268402300609392851777, 2.80489143957500026839932213363, 3.84198284398957982147077015280, 4.92948800021764153561173002148, 5.81684998270979679375964446625, 6.58127442055788450585742894483, 7.87641588695788242097833597140, 8.78288132581306100252795239665, 9.23533922148352663940396199508, 10.1007115692366831903248610807, 11.32088081027547407097443258232, 11.92535173073657130967283756036, 12.61596323900256717467363096612, 13.4322395081298988486865867250, 13.97022388316886448129243405969, 14.43389602957685768457105208180, 15.84018489082161828235509151985, 16.94156766092078863129179509717, 17.58107370884539969018012982812, 18.40151945485657290776184155948, 18.881135031529046686806361195633, 20.004999269288165951778422595551, 20.25228218706301473117760355807

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