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Properties

Label 1-1161-1161.182-r1-0-0
Degree $1$
Conductor $1161$
Sign $-0.681 - 0.731i$
Analytic cond. $124.766$
Root an. cond. $124.766$
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  + (0.411 − 0.911i)2-s + (−0.661 − 0.749i)4-s + (0.797 − 0.603i)5-s + (−0.939 − 0.342i)7-s + (−0.955 + 0.294i)8-s + (−0.222 − 0.974i)10-s + (0.318 + 0.947i)11-s + (−0.124 − 0.992i)13-s + (−0.698 + 0.715i)14-s + (−0.124 + 0.992i)16-s + (0.733 − 0.680i)17-s + (0.365 + 0.930i)19-s + (−0.980 − 0.198i)20-s + (0.995 + 0.0995i)22-s + (0.853 + 0.521i)23-s + ⋯
L(s)  = 1  + (0.411 − 0.911i)2-s + (−0.661 − 0.749i)4-s + (0.797 − 0.603i)5-s + (−0.939 − 0.342i)7-s + (−0.955 + 0.294i)8-s + (−0.222 − 0.974i)10-s + (0.318 + 0.947i)11-s + (−0.124 − 0.992i)13-s + (−0.698 + 0.715i)14-s + (−0.124 + 0.992i)16-s + (0.733 − 0.680i)17-s + (0.365 + 0.930i)19-s + (−0.980 − 0.198i)20-s + (0.995 + 0.0995i)22-s + (0.853 + 0.521i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1161\)    =    \(3^{3} \cdot 43\)
Sign: $-0.681 - 0.731i$
Analytic conductor: \(124.766\)
Root analytic conductor: \(124.766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1161} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1161,\ (1:\ ),\ -0.681 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.118347277 - 2.571779851i\)
\(L(\frac12)\) \(\approx\) \(1.118347277 - 2.571779851i\)
\(L(1)\) \(\approx\) \(1.073923767 - 0.8867152125i\)
\(L(1)\) \(\approx\) \(1.073923767 - 0.8867152125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.411 - 0.911i)T \)
5 \( 1 + (0.797 - 0.603i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.318 + 0.947i)T \)
13 \( 1 + (-0.124 - 0.992i)T \)
17 \( 1 + (0.733 - 0.680i)T \)
19 \( 1 + (0.365 + 0.930i)T \)
23 \( 1 + (0.853 + 0.521i)T \)
29 \( 1 + (0.583 + 0.811i)T \)
31 \( 1 + (0.698 - 0.715i)T \)
37 \( 1 + T \)
41 \( 1 + (0.969 + 0.246i)T \)
47 \( 1 + (-0.878 - 0.478i)T \)
53 \( 1 + (0.733 - 0.680i)T \)
59 \( 1 + (-0.542 - 0.840i)T \)
61 \( 1 + (0.698 + 0.715i)T \)
67 \( 1 + (-0.661 - 0.749i)T \)
71 \( 1 + (0.988 + 0.149i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.698 - 0.715i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 + (-0.318 - 0.947i)T \)
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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

−21.43695523886486087013710223014, −21.20527186693727135206002688719, −19.480286766155373479718964801389, −18.9839189645378646439771155360, −18.210048918380812706059097159966, −17.285494743022272981329331390622, −16.66436910795469074419354873221, −16.00290353260363599121228474467, −15.08828253986815292767762973308, −14.33966389118903491509254176117, −13.69483026258305468973037763722, −13.06353487390878705076513178894, −12.149183818621149567172681447420, −11.213622705273933736032399606298, −10.06848273643000240504754179863, −9.26374349507856133643084013700, −8.71308653168027583578863885119, −7.5025155408855808998347454084, −6.49426933464737820676053308234, −6.28354843060668333143770820569, −5.358566703258296477672501491985, −4.25502670847497970243951230106, −3.17697695974626931972625979854, −2.60615186890200447741752991039, −0.88582582261409294357047258427, 0.647332428171804428336736319340, 1.353260102212949441339788007660, 2.56338192718965986566187424593, 3.30447965690272205479036291256, 4.36280808815042654495396448776, 5.25677016346433741508386289085, 5.90985414150009103914798827633, 6.95252117763318763974212271237, 8.14326198252327391720338126459, 9.337195902957330190786282387243, 9.82567525595364600118013241652, 10.24823321790940372444510171144, 11.45599538347659570091062586910, 12.49527518959585782423745243542, 12.74839994036863716593660917478, 13.57472316771399186522005687833, 14.33097225814780379693420848314, 15.16883386581164035894305781163, 16.22119758979852892870837911858, 17.04066261150691874000999768283, 17.8579165067893332040431379690, 18.515981178293588853357513753719, 19.59541760910764107784382907306, 20.08788681426646198096460681190, 20.77351378990727787270213873000

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