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Properties

Label 1-1157-1157.1005-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.179 - 0.983i$
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  + (−0.0475 + 0.998i)2-s + (0.771 + 0.636i)3-s + (−0.995 − 0.0950i)4-s + (−0.989 − 0.142i)5-s + (−0.672 + 0.739i)6-s + (−0.118 − 0.992i)7-s + (0.142 − 0.989i)8-s + (0.189 + 0.981i)9-s + (0.189 − 0.981i)10-s + (−0.786 + 0.618i)11-s + (−0.707 − 0.707i)12-s + (0.997 − 0.0713i)14-s + (−0.672 − 0.739i)15-s + (0.981 + 0.189i)16-s + (0.998 − 0.0475i)17-s + (−0.989 + 0.142i)18-s + ⋯
L(s)  = 1  + (−0.0475 + 0.998i)2-s + (0.771 + 0.636i)3-s + (−0.995 − 0.0950i)4-s + (−0.989 − 0.142i)5-s + (−0.672 + 0.739i)6-s + (−0.118 − 0.992i)7-s + (0.142 − 0.989i)8-s + (0.189 + 0.981i)9-s + (0.189 − 0.981i)10-s + (−0.786 + 0.618i)11-s + (−0.707 − 0.707i)12-s + (0.997 − 0.0713i)14-s + (−0.672 − 0.739i)15-s + (0.981 + 0.189i)16-s + (0.998 − 0.0475i)17-s + (−0.989 + 0.142i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.179 - 0.983i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.179 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2563168258 + 0.3071784206i\)
\(L(\frac12)\) \(\approx\) \(-0.2563168258 + 0.3071784206i\)
\(L(1)\) \(\approx\) \(0.6823511338 + 0.5479440240i\)
\(L(1)\) \(\approx\) \(0.6823511338 + 0.5479440240i\)

Euler product

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

−20.45740749687765324943788678702, −19.562657270976899883013657263551, −19.05993381605015151046081247154, −18.61469265864683634716473620662, −17.95347661946946256281027013668, −16.76038134141583485951173574356, −15.609704547926752267277316306975, −14.9683962745433482304352087893, −14.23303596599731308665988752802, −13.104333953526859880658195797872, −12.78291518682540710378146196665, −11.84305467688658741683222534593, −11.322338174205892535595102609850, −10.2884356563966036350601305703, −9.22740381310428208957321638665, −8.56820701136710765893048272291, −8.00938004129624230750382714989, −7.089716339176205745512248800086, −5.783126415704654746169214600168, −4.805304064615186469556624241668, −3.59742529986788955819455213048, −2.99628103687143033641416877855, −2.33656234828578610100786948090, −1.06174596525568282191063179785, −0.08947940629776748406097390282, 1.19609850533302518630161581673, 2.98317142163812192776271373921, 3.78516972671315475850142913132, 4.54700781086719904045487551116, 5.11657204846839064227962249953, 6.56729595496876650973377316306, 7.47416980454727617935951547444, 7.99421647264800104089132483601, 8.55654846089434919698529382642, 9.862417122873817893717953079616, 10.14532311201886080471102100182, 11.23149050399276537566885735990, 12.69773901022675342925105925659, 13.09382686855357642230359584339, 14.33253677759438309413935610637, 14.612502811668953729185358763673, 15.52267562920940856644892791131, 16.15051135580569538815000949751, 16.6858770293470910376860726909, 17.53429585442382628663746730303, 18.82992738982170536857317336855, 19.13252521128389687852262677318, 20.16272660064947750787378596640, 20.775010107111433892444983110979, 21.58239651836296215634681970016

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