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Properties

Label 1-1157-1157.21-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.000775 + 0.999i$
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  + (0.540 + 0.841i)2-s + (0.909 − 0.415i)3-s + (−0.415 + 0.909i)4-s + (−0.989 + 0.142i)5-s + (0.841 + 0.540i)6-s + (0.142 + 0.989i)7-s + (−0.989 + 0.142i)8-s + (0.654 − 0.755i)9-s + (−0.654 − 0.755i)10-s + (0.989 + 0.142i)11-s + i·12-s + (−0.755 + 0.654i)14-s + (−0.841 + 0.540i)15-s + (−0.654 − 0.755i)16-s + (0.841 + 0.540i)17-s + (0.989 + 0.142i)18-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)2-s + (0.909 − 0.415i)3-s + (−0.415 + 0.909i)4-s + (−0.989 + 0.142i)5-s + (0.841 + 0.540i)6-s + (0.142 + 0.989i)7-s + (−0.989 + 0.142i)8-s + (0.654 − 0.755i)9-s + (−0.654 − 0.755i)10-s + (0.989 + 0.142i)11-s + i·12-s + (−0.755 + 0.654i)14-s + (−0.841 + 0.540i)15-s + (−0.654 − 0.755i)16-s + (0.841 + 0.540i)17-s + (0.989 + 0.142i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.000775 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.000775 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.000775 + 0.999i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.000775 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.754043249 + 2.751909384i\)
\(L(\frac12)\) \(\approx\) \(2.754043249 + 2.751909384i\)
\(L(1)\) \(\approx\) \(1.571105324 + 0.8599445829i\)
\(L(1)\) \(\approx\) \(1.571105324 + 0.8599445829i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.540 + 0.841i)T \)
3 \( 1 + (0.909 - 0.415i)T \)
5 \( 1 + (-0.989 + 0.142i)T \)
7 \( 1 + (0.142 + 0.989i)T \)
11 \( 1 + (0.989 + 0.142i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (-0.755 - 0.654i)T \)
29 \( 1 + (0.989 - 0.142i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (0.909 + 0.415i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (-0.281 + 0.959i)T \)
67 \( 1 + (-0.909 + 0.415i)T \)
71 \( 1 + (0.989 + 0.142i)T \)
73 \( 1 + (-0.755 + 0.654i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.989 - 0.142i)T \)
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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

−20.68673214395835626113174268490, −20.09390158986941747337034685399, −19.714598225742622398056815177931, −18.98764636901778826442697573504, −18.16831283424693871329573816772, −16.79437771040422282480037865789, −16.13116349491809760258407184084, −15.21301339333081671800491303795, −14.43056640674435303068794872513, −13.962585329591196714481443485719, −13.21778966157400120640766566458, −12.06012454430290557768693670522, −11.62654948761788230049902974879, −10.5952958440019847864751467288, −9.87330221398965540346404578537, −9.18635941394486828634242219744, −8.06680015702886732051263763390, −7.5009529497392925636332608756, −6.25544566158279096124448800699, −4.8903513071295273240155191032, −4.21840218312559049332054990076, −3.59777776503779004313492460207, −2.96217866548361915338463857385, −1.516822891177174216958879971593, −0.771954015747703966538257064469, 0.90151278269418175986430920024, 2.42405734613509418168980394680, 3.24351202849063921551731098552, 4.01890771406206579886368512404, 4.88064314562223316941294655122, 6.13867190990805880691464023505, 6.81602827296273310352728022947, 7.68166647112324712295665892083, 8.39282743487197185201948582074, 8.87976590985704264516029897565, 9.88080980155714583558244164877, 11.50774193351083518075260586963, 12.204560778033205373592432029093, 12.54071181173067313163796501594, 13.76120548279671418334766618198, 14.45864990070905286591738783802, 14.94726232806993737316539017916, 15.68807991735889777699810137483, 16.26679297280342438549507482670, 17.48191585100426777705335599076, 18.20409726270182369760409382453, 18.995120552561862972275725098, 19.65663871116246096269585357233, 20.50355114853954836357091436675, 21.43535187954562332287958013840

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