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Properties

Label 1-1137-1137.167-r1-0-0
Degree $1$
Conductor $1137$
Sign $0.786 + 0.617i$
Analytic cond. $122.187$
Root an. cond. $122.187$
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  + (0.797 + 0.603i)2-s + (0.270 + 0.962i)4-s + (−0.623 + 0.781i)5-s + (−0.998 + 0.0498i)7-s + (−0.365 + 0.930i)8-s + (−0.969 + 0.246i)10-s + (−0.766 + 0.642i)11-s + (0.270 − 0.962i)13-s + (−0.826 − 0.563i)14-s + (−0.853 + 0.521i)16-s + (0.661 − 0.749i)17-s + (0.921 − 0.388i)19-s + (−0.921 − 0.388i)20-s + (−0.998 + 0.0498i)22-s + (0.988 − 0.149i)23-s + ⋯
L(s)  = 1  + (0.797 + 0.603i)2-s + (0.270 + 0.962i)4-s + (−0.623 + 0.781i)5-s + (−0.998 + 0.0498i)7-s + (−0.365 + 0.930i)8-s + (−0.969 + 0.246i)10-s + (−0.766 + 0.642i)11-s + (0.270 − 0.962i)13-s + (−0.826 − 0.563i)14-s + (−0.853 + 0.521i)16-s + (0.661 − 0.749i)17-s + (0.921 − 0.388i)19-s + (−0.921 − 0.388i)20-s + (−0.998 + 0.0498i)22-s + (0.988 − 0.149i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1137\)    =    \(3 \cdot 379\)
Sign: $0.786 + 0.617i$
Analytic conductor: \(122.187\)
Root analytic conductor: \(122.187\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1137} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1137,\ (1:\ ),\ 0.786 + 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.866249927 + 0.6447720066i\)
\(L(\frac12)\) \(\approx\) \(1.866249927 + 0.6447720066i\)
\(L(1)\) \(\approx\) \(1.109220561 + 0.5696096034i\)
\(L(1)\) \(\approx\) \(1.109220561 + 0.5696096034i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
379 \( 1 \)
good2 \( 1 + (0.797 + 0.603i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + (-0.998 + 0.0498i)T \)
11 \( 1 + (-0.766 + 0.642i)T \)
13 \( 1 + (0.270 - 0.962i)T \)
17 \( 1 + (0.661 - 0.749i)T \)
19 \( 1 + (0.921 - 0.388i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (-0.365 + 0.930i)T \)
31 \( 1 + (-0.583 - 0.811i)T \)
37 \( 1 + (0.365 - 0.930i)T \)
41 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (-0.969 - 0.246i)T \)
47 \( 1 + (-0.456 + 0.889i)T \)
53 \( 1 + (0.411 - 0.911i)T \)
59 \( 1 + (0.988 - 0.149i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.0747 + 0.997i)T \)
71 \( 1 + (0.853 + 0.521i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.733 + 0.680i)T \)
89 \( 1 + (0.124 - 0.992i)T \)
97 \( 1 + (0.921 + 0.388i)T \)
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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

−21.21435726909189356602834720930, −20.27972379821495654065333997133, −19.63921995845738002547744842317, −18.92597629886915275888866578579, −18.44351779014085326683556528192, −16.66547439737028576083866993529, −16.45056689069701299867337346901, −15.53351309416264292479816243126, −14.85604214330135450926928658774, −13.5679960396153637689389269052, −13.32055806746683624971725527808, −12.38953157029534908704910075877, −11.77736503709881916926233487715, −10.968375737478346140793157079038, −9.99373057474498886795511994306, −9.28779368146247173858722317908, −8.31353537201735093797078566577, −7.2275182817801794209093697677, −6.227573859536729616926147305914, −5.435514564280984900833138515803, −4.591251931127127423747903243940, −3.534282515324517346269449422529, −3.14361450647274147607416768325, −1.66903002801842841813493630931, −0.73390203077547156219523632280, 0.419463254745772590905411898573, 2.49584314543804699548571246030, 3.15229462722150864166744551159, 3.75682516236677337650847630235, 5.07306677007577739357248982769, 5.65798001476081000340445203336, 6.86304206035434510300296807303, 7.275354686118104903275369118379, 8.01607388617060914888566476025, 9.207348503820474190382027507986, 10.20741606624680756795158786569, 11.08490207444016130180471053100, 11.93795898823417267660129352559, 12.83363532439079115596434666769, 13.25007215119563279797012868601, 14.37818981125151895792012016831, 15.00300167661310482979940228025, 15.84643770530600938367863718926, 16.07900676283915560943261255967, 17.22182805200234083462716598278, 18.21491085225715069059691651009, 18.67981292897874072789748201161, 19.94384153544402587250857654341, 20.3967979891747802204740174728, 21.38727480810617373951793431608

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