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Properties

Label 1-1053-1053.238-r0-0-0
Degree $1$
Conductor $1053$
Sign $0.928 + 0.372i$
Analytic cond. $4.89011$
Root an. cond. $4.89011$
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  + (0.835 + 0.549i)2-s + (0.396 + 0.918i)4-s + (0.286 − 0.957i)5-s + (−0.597 − 0.802i)7-s + (−0.173 + 0.984i)8-s + (0.766 − 0.642i)10-s + (0.286 + 0.957i)11-s + (−0.0581 − 0.998i)14-s + (−0.686 + 0.727i)16-s + (0.766 − 0.642i)17-s + (−0.173 + 0.984i)19-s + (0.993 − 0.116i)20-s + (−0.286 + 0.957i)22-s + (0.597 − 0.802i)23-s + (−0.835 − 0.549i)25-s + ⋯
L(s)  = 1  + (0.835 + 0.549i)2-s + (0.396 + 0.918i)4-s + (0.286 − 0.957i)5-s + (−0.597 − 0.802i)7-s + (−0.173 + 0.984i)8-s + (0.766 − 0.642i)10-s + (0.286 + 0.957i)11-s + (−0.0581 − 0.998i)14-s + (−0.686 + 0.727i)16-s + (0.766 − 0.642i)17-s + (−0.173 + 0.984i)19-s + (0.993 − 0.116i)20-s + (−0.286 + 0.957i)22-s + (0.597 − 0.802i)23-s + (−0.835 − 0.549i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.928 + 0.372i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ 0.928 + 0.372i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.525841097 + 0.4874486014i\)
\(L(\frac12)\) \(\approx\) \(2.525841097 + 0.4874486014i\)
\(L(1)\) \(\approx\) \(1.718117309 + 0.3207937665i\)
\(L(1)\) \(\approx\) \(1.718117309 + 0.3207937665i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.835 + 0.549i)T \)
5 \( 1 + (0.286 - 0.957i)T \)
7 \( 1 + (-0.597 - 0.802i)T \)
11 \( 1 + (0.286 + 0.957i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.597 - 0.802i)T \)
29 \( 1 + (0.893 - 0.448i)T \)
31 \( 1 + (0.993 + 0.116i)T \)
37 \( 1 + (0.939 + 0.342i)T \)
41 \( 1 + (0.0581 + 0.998i)T \)
43 \( 1 + (-0.286 - 0.957i)T \)
47 \( 1 + (0.993 - 0.116i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.286 - 0.957i)T \)
61 \( 1 + (-0.993 + 0.116i)T \)
67 \( 1 + (0.835 - 0.549i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (0.893 - 0.448i)T \)
83 \( 1 + (-0.893 + 0.448i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.973 - 0.230i)T \)
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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

−21.67390598709424377558861267537, −21.09528589598417896844742658655, −19.703699301157259229425006003246, −19.25398599307435878124861965933, −18.70427936201941065291841274880, −17.785783511776343488978202261141, −16.66448577327147756515235217645, −15.59625794689280897414075362697, −15.178953524440634419755522773760, −14.23264643875228311766390486918, −13.639374762440377986297862800630, −12.80225533156443904123320893269, −11.93721332411664468408151872238, −11.19505373542122117339545788247, −10.49994054913024059866706324554, −9.62471453957274016388614816170, −8.83639214561929754302229424762, −7.44558131651180090597528175155, −6.3507614021180708659565790452, −6.033663646211443973868505949914, −5.05809127232933787482082681812, −3.76321388911063762979416788528, −3.01284029248120937230195891854, −2.45206915957581111478291903888, −1.1008313145829288635774694668, 1.01239637071745075526012927585, 2.32845659961311490792029483044, 3.46494343446765923813807134147, 4.41718862952213921092936391049, 4.92598639250100152960482605271, 6.06171981125898719908758910365, 6.75174781603647924172675498924, 7.696551253438326548123374667454, 8.43717525922726625729208346724, 9.60113122518278251440495283346, 10.21426414259401342542273992329, 11.5793221771913667743164836545, 12.430602076830698038189969870088, 12.80480508081589209958273161184, 13.78649158283666987322895543404, 14.29504923921563432037134180203, 15.34815129434049484599134350010, 16.13040200530655701223640194510, 16.89525988854754120056852488937, 17.152442301953051144334694222527, 18.28753080740992211540948846183, 19.502570345593238975751097312512, 20.421858620341867825536467366302, 20.68563326449464932123683169384, 21.57281090663734858068537397132

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