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Properties

Label 1-1157-1157.101-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.0878 - 0.996i$
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  + (0.5 − 0.866i)2-s + (−0.258 − 0.965i)3-s + (−0.5 − 0.866i)4-s − i·5-s + (−0.965 − 0.258i)6-s + (−0.965 − 0.258i)7-s − 8-s + (−0.866 + 0.5i)9-s + (−0.866 − 0.5i)10-s + (−0.5 + 0.866i)11-s + (−0.707 + 0.707i)12-s + (−0.707 + 0.707i)14-s + (−0.965 + 0.258i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + i·18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.258 − 0.965i)3-s + (−0.5 − 0.866i)4-s − i·5-s + (−0.965 − 0.258i)6-s + (−0.965 − 0.258i)7-s − 8-s + (−0.866 + 0.5i)9-s + (−0.866 − 0.5i)10-s + (−0.5 + 0.866i)11-s + (−0.707 + 0.707i)12-s + (−0.707 + 0.707i)14-s + (−0.965 + 0.258i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0878 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0878 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.0878 - 0.996i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.0878 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5900973379 - 0.5403388051i\)
\(L(\frac12)\) \(\approx\) \(0.5900973379 - 0.5403388051i\)
\(L(1)\) \(\approx\) \(0.4936042412 - 0.6899744871i\)
\(L(1)\) \(\approx\) \(0.4936042412 - 0.6899744871i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 - iT \)
7 \( 1 + (-0.965 - 0.258i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (-0.258 - 0.965i)T \)
29 \( 1 + (-0.965 + 0.258i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + (-0.258 - 0.965i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 - iT \)
53 \( 1 + iT \)
59 \( 1 + (-0.258 + 0.965i)T \)
61 \( 1 + (-0.965 - 0.258i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + T \)
79 \( 1 - iT \)
83 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−21.76624249818341693948970892446, −20.917491969126335480693239916079, −19.651868494263933059498875166728, −18.83649801154475937128817089607, −18.05394512228113376274129095772, −17.08489172341379841185623909440, −16.50310038389927389140784920655, −15.65137197891831462175945097485, −15.25238566096232817236326292298, −14.506772761348295202212971583251, −13.5925725035741835227350183318, −12.93837544373067344045210324445, −11.684231444324978085566168192471, −11.153630558177327965754359194042, −9.96256105405628580723932030456, −9.50484944268600453113670011340, −8.38594409661291955770300716009, −7.55608207169104393554573507993, −6.346271941163811990275654715429, −6.03600086698149408316611595418, −5.15806766843812415109621273117, −3.98519429647287611786894876685, −3.26134736458150349032554714857, −2.73776968646368328895496600266, −0.22522477251666166572680497047, 0.61142542446046326737855617572, 1.55938265937133717743969905079, 2.43892068756175739106040468804, 3.49119579381831044270061653332, 4.54622942670949946730969095635, 5.42425948064768977908022804816, 6.10977185437632578256264980069, 7.15310309191500875427411672496, 8.13333035536935776505262956121, 9.09988301868499395924987991323, 9.97357598390302356718588218056, 10.63122366669398606949614588151, 12.00177665179018415240627926713, 12.269950425073395754803822408844, 12.89939176773310776256436077692, 13.54122477453333222109135356742, 14.31385332591552468343979981794, 15.3881406271969535072544381754, 16.48702185001431753940124713465, 16.99587426259712762440839497330, 18.19527861623218667633075953133, 18.641459974384912515021319898692, 19.50477673076838523316969498283, 20.23319497348718419449713076726, 20.60773033252225255595334196519

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