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Properties

Label 1-1157-1157.154-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.999 - 0.0383i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (−0.998 + 0.0475i)2-s + (−0.771 + 0.636i)3-s + (0.995 − 0.0950i)4-s + (0.142 + 0.989i)5-s + (0.739 − 0.672i)6-s + (0.992 + 0.118i)7-s + (−0.989 + 0.142i)8-s + (0.189 − 0.981i)9-s + (−0.189 − 0.981i)10-s + (0.618 − 0.786i)11-s + (−0.707 + 0.707i)12-s + (−0.997 − 0.0713i)14-s + (−0.739 − 0.672i)15-s + (0.981 − 0.189i)16-s + (−0.998 − 0.0475i)17-s + (−0.142 + 0.989i)18-s + ⋯
L(s)  = 1  + (−0.998 + 0.0475i)2-s + (−0.771 + 0.636i)3-s + (0.995 − 0.0950i)4-s + (0.142 + 0.989i)5-s + (0.739 − 0.672i)6-s + (0.992 + 0.118i)7-s + (−0.989 + 0.142i)8-s + (0.189 − 0.981i)9-s + (−0.189 − 0.981i)10-s + (0.618 − 0.786i)11-s + (−0.707 + 0.707i)12-s + (−0.997 − 0.0713i)14-s + (−0.739 − 0.672i)15-s + (0.981 − 0.189i)16-s + (−0.998 − 0.0475i)17-s + (−0.142 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.999 - 0.0383i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.999 - 0.0383i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8278287574 + 0.01589318561i\)
\(L(\frac12)\) \(\approx\) \(0.8278287574 + 0.01589318561i\)
\(L(1)\) \(\approx\) \(0.6568651957 + 0.1238648393i\)
\(L(1)\) \(\approx\) \(0.6568651957 + 0.1238648393i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.998 + 0.0475i)T \)
3 \( 1 + (-0.771 + 0.636i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (0.992 + 0.118i)T \)
11 \( 1 + (0.618 - 0.786i)T \)
17 \( 1 + (-0.998 - 0.0475i)T \)
19 \( 1 + (0.828 - 0.560i)T \)
23 \( 1 + (0.899 + 0.436i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (0.997 + 0.0713i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (0.165 - 0.986i)T \)
43 \( 1 + (0.393 - 0.919i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (0.636 - 0.771i)T \)
61 \( 1 + (-0.853 - 0.520i)T \)
67 \( 1 + (0.0950 - 0.995i)T \)
71 \( 1 + (-0.928 + 0.371i)T \)
73 \( 1 + (0.755 - 0.654i)T \)
79 \( 1 + (-0.755 + 0.654i)T \)
83 \( 1 + (-0.212 - 0.977i)T \)
97 \( 1 + (-0.618 - 0.786i)T \)
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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

−20.98071064609430292008585263613, −20.397988788612663826645045241482, −19.725686978238745298603938541547, −18.833524950843840185481099096838, −17.853287457375596618202346964333, −17.62374583783823780436564854555, −16.871623549632146875376831614996, −16.274189023271097804843403001939, −15.31380926069139472803672764554, −14.326293803967056297792208553509, −13.19711207678633691298755386640, −12.41844292730412675970960544961, −11.72856092878977033967210099932, −11.1649643416357396312413097038, −10.2072714644323370428260274736, −9.2892644992532244709023664257, −8.46282597086585116734609190542, −7.739346998796337298739621757578, −6.95133198142235269153269928980, −6.09175257168023773678704467800, −5.058682578844222597433191703515, −4.37104934556489880947097098389, −2.56618861417735179380202149718, −1.37845399258560951955944364292, −1.22931268844033018178159796361, 0.60468393662140976210125363796, 1.855991372222093062970275718297, 2.956549298750100427798473539963, 3.96463957722999665346895915790, 5.238426690803932068967417339644, 6.025470064865281098333312960813, 6.83395417947501410278094648750, 7.56162323425137459156694648954, 8.77905426399427898937389913372, 9.33553890658122193134138999843, 10.33699954594693004210372236285, 11.11937326704757578711337746968, 11.32040879962721329812479701944, 12.10815904411901925767516576066, 13.70880213457190339714549431383, 14.51568628595780529850880548440, 15.53882318842070714457225463999, 15.62387379806862482315529253620, 17.06791719814412269620172304664, 17.3366212329302781805765873159, 18.0570830338667870344480226818, 18.76235318640057507250043122568, 19.557246715831213896002676352202, 20.58769738442374078297896506229, 21.2809375474525994801874384047

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