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Properties

Label 1-1183-1183.229-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.290 + 0.956i$
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  + (−0.979 − 0.200i)2-s + (0.632 + 0.774i)3-s + (0.919 + 0.391i)4-s + (0.960 − 0.278i)5-s + (−0.464 − 0.885i)6-s + (−0.822 − 0.568i)8-s + (−0.200 + 0.979i)9-s + (−0.996 + 0.0804i)10-s + (−0.979 + 0.200i)11-s + (0.278 + 0.960i)12-s + (0.822 + 0.568i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (0.391 − 0.919i)18-s + (−0.866 + 0.5i)19-s + (0.992 + 0.120i)20-s + ⋯
L(s)  = 1  + (−0.979 − 0.200i)2-s + (0.632 + 0.774i)3-s + (0.919 + 0.391i)4-s + (0.960 − 0.278i)5-s + (−0.464 − 0.885i)6-s + (−0.822 − 0.568i)8-s + (−0.200 + 0.979i)9-s + (−0.996 + 0.0804i)10-s + (−0.979 + 0.200i)11-s + (0.278 + 0.960i)12-s + (0.822 + 0.568i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (0.391 − 0.919i)18-s + (−0.866 + 0.5i)19-s + (0.992 + 0.120i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.290 + 0.956i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6935178942 + 0.9351816256i\)
\(L(\frac12)\) \(\approx\) \(0.6935178942 + 0.9351816256i\)
\(L(1)\) \(\approx\) \(0.8655250856 + 0.3114029302i\)
\(L(1)\) \(\approx\) \(0.8655250856 + 0.3114029302i\)

Euler product

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

−21.003085254014372548313344882678, −20.11718383899470900726811397542, −19.15073710345918282474527968825, −18.65655919795795181537797702660, −18.01240048322287308157736882997, −17.43699046381188508901040344649, −16.54008938320958263830721491775, −15.62762593932271939478598554415, −14.70012150631600637084612128713, −14.165729148458211692218005378865, −13.16531689911726895185481990805, −12.55509575077238663144398609714, −11.3522060262134577984759972408, −10.58364322449551493662438235141, −9.7447016825941111649943579552, −9.00967068692642053407213238602, −8.3041943043267174428436398064, −7.402981705246080710116300364493, −6.73805992947587597162784634744, −5.974287006742397045367786463573, −5.0179883836224062288241121152, −3.11694142947119225236687302406, −2.53144164107653881390652973720, −1.74791462609236293522334673024, −0.55986663724272506530532089017, 1.446139843677075326059020063656, 2.26747940649514136494282342631, 3.0583583159386590296778174777, 4.15711502851017713264950732278, 5.348530729422678651032259760625, 6.10237967638132766521829609020, 7.3477738646276560913875958360, 8.24103932394454415839677563291, 8.75805979064862248749430452688, 9.7330686332708956838934098285, 10.21060822163395876620964112634, 10.74531539019166075238159437296, 11.90780610038894958401879967044, 13.013266472392860343199800941555, 13.52086147043330250731438938496, 14.82509088589756282070674580210, 15.2542340984883656452459815752, 16.21889775883652162823100536793, 16.95487338899008848650601323001, 17.4353256966020494875803584074, 18.493380145382154816614693972, 19.136959524554800451948946995661, 19.94135871534260003228423207192, 20.82866234253791159573677342559, 21.19433133019909897127737265404

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