Properties

Label 1-1157-1157.29-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.983 + 0.178i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (0.928 + 0.371i)2-s + (0.999 + 0.0237i)3-s + (0.723 + 0.690i)4-s + (0.909 − 0.415i)5-s + (0.919 + 0.393i)6-s + (0.986 − 0.165i)7-s + (0.415 + 0.909i)8-s + (0.998 + 0.0475i)9-s + (0.998 − 0.0475i)10-s + (−0.580 − 0.814i)11-s + (0.707 + 0.707i)12-s + (0.977 + 0.212i)14-s + (0.919 − 0.393i)15-s + (0.0475 + 0.998i)16-s + (−0.371 − 0.928i)17-s + (0.909 + 0.415i)18-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)2-s + (0.999 + 0.0237i)3-s + (0.723 + 0.690i)4-s + (0.909 − 0.415i)5-s + (0.919 + 0.393i)6-s + (0.986 − 0.165i)7-s + (0.415 + 0.909i)8-s + (0.998 + 0.0475i)9-s + (0.998 − 0.0475i)10-s + (−0.580 − 0.814i)11-s + (0.707 + 0.707i)12-s + (0.977 + 0.212i)14-s + (0.919 − 0.393i)15-s + (0.0475 + 0.998i)16-s + (−0.371 − 0.928i)17-s + (0.909 + 0.415i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.983 + 0.178i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.983 + 0.178i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(8.853571593 + 0.7967514582i\)
\(L(\frac12)\) \(\approx\) \(8.853571593 + 0.7967514582i\)
\(L(1)\) \(\approx\) \(3.425059825 + 0.4196580526i\)
\(L(1)\) \(\approx\) \(3.425059825 + 0.4196580526i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.928 + 0.371i)T \)
3 \( 1 + (0.999 + 0.0237i)T \)
5 \( 1 + (0.909 - 0.415i)T \)
7 \( 1 + (0.986 - 0.165i)T \)
11 \( 1 + (-0.580 - 0.814i)T \)
17 \( 1 + (-0.371 - 0.928i)T \)
19 \( 1 + (0.672 + 0.739i)T \)
23 \( 1 + (-0.952 - 0.304i)T \)
29 \( 1 + (0.771 - 0.636i)T \)
31 \( 1 + (0.212 - 0.977i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (-0.853 + 0.520i)T \)
43 \( 1 + (0.771 + 0.636i)T \)
47 \( 1 + (0.281 + 0.959i)T \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 + (-0.999 + 0.0237i)T \)
61 \( 1 + (0.899 - 0.436i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (0.0950 + 0.995i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (0.540 + 0.841i)T \)
83 \( 1 + (0.800 - 0.599i)T \)
97 \( 1 + (0.580 - 0.814i)T \)
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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

−21.04756327492907949136256304014, −20.471067192360328442116347004735, −19.84257669019612903199105634917, −18.88695441328256767079029501959, −18.04214901516055320039593524707, −17.502207142753460122569277525186, −15.98823592028034726595885911725, −15.2358601860157698247414235412, −14.74587034447361773325446578042, −13.8961070601748169544593273326, −13.551998918037739576901766918078, −12.603397888741330213079750465181, −11.83274792309365451560528941976, −10.55354446576862325491734699274, −10.28138633915756323734195613213, −9.2483458287846754924461433692, −8.258618214673128728849189250480, −7.24465647731689958126747270833, −6.56377827239796488706241339734, −5.32017062406873115293349582185, −4.75953881766219190702654955208, −3.677178237646204732338075209348, −2.69982247300167610538512156928, −1.99613747197766507129225332168, −1.40015645118779623982499146157, 1.12083333205327107618229905319, 2.20161884892136794953446672924, 2.781404101230192669500009924117, 3.98783973113224276840044514820, 4.75379998354762671788115381746, 5.566757914069796300706773639941, 6.433852087091618017029764323446, 7.69549231321871123316981800336, 8.05253629506759908136327132232, 8.98621238204019997217767340283, 9.984033509635454840567423066067, 10.91499164835957699833833196203, 11.90197686005172967110841022276, 12.8014969030353140389255102714, 13.68019428087088614590873832647, 13.98508115125277727397908638600, 14.534843252884247285109709144133, 15.698196982133443989026313564167, 16.13401974699585933298472372265, 17.142803942164268098744398856940, 17.999352179828566231566079467875, 18.719352716030734228859066430495, 20.094542157821538366052990233192, 20.51008076682337501208780398472, 21.1571818348231353860309727285

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