Properties

Label 1-1053-1053.238-r0-0-0
Degree 11
Conductor 10531053
Sign 0.928+0.372i0.928 + 0.372i
Analytic cond. 4.890114.89011
Root an. cond. 4.890114.89011
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

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

Λ(s)=(1053s/2ΓR(s)L(s)=((0.928+0.372i)Λ(1s)\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}
Λ(s)=(1053s/2ΓR(s)L(s)=((0.928+0.372i)Λ(1s)\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: 11
Conductor: 10531053    =    34133^{4} \cdot 13
Sign: 0.928+0.372i0.928 + 0.372i
Analytic conductor: 4.890114.89011
Root analytic conductor: 4.890114.89011
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1053(238,)\chi_{1053} (238, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1053, (0: ), 0.928+0.372i)(1,\ 1053,\ (0:\ ),\ 0.928 + 0.372i)

Particular Values

L(12)L(\frac{1}{2}) \approx 2.525841097+0.4874486014i2.525841097 + 0.4874486014i
L(12)L(\frac12) \approx 2.525841097+0.4874486014i2.525841097 + 0.4874486014i
L(1)L(1) \approx 1.718117309+0.3207937665i1.718117309 + 0.3207937665i
L(1)L(1) \approx 1.718117309+0.3207937665i1.718117309 + 0.3207937665i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
13 1 1
good2 1+(0.835+0.549i)T 1 + (0.835 + 0.549i)T
5 1+(0.2860.957i)T 1 + (0.286 - 0.957i)T
7 1+(0.5970.802i)T 1 + (-0.597 - 0.802i)T
11 1+(0.286+0.957i)T 1 + (0.286 + 0.957i)T
17 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
19 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
23 1+(0.5970.802i)T 1 + (0.597 - 0.802i)T
29 1+(0.8930.448i)T 1 + (0.893 - 0.448i)T
31 1+(0.993+0.116i)T 1 + (0.993 + 0.116i)T
37 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
41 1+(0.0581+0.998i)T 1 + (0.0581 + 0.998i)T
43 1+(0.2860.957i)T 1 + (-0.286 - 0.957i)T
47 1+(0.9930.116i)T 1 + (0.993 - 0.116i)T
53 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
59 1+(0.2860.957i)T 1 + (0.286 - 0.957i)T
61 1+(0.993+0.116i)T 1 + (-0.993 + 0.116i)T
67 1+(0.8350.549i)T 1 + (0.835 - 0.549i)T
71 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
73 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
79 1+(0.8930.448i)T 1 + (0.893 - 0.448i)T
83 1+(0.893+0.448i)T 1 + (-0.893 + 0.448i)T
89 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
97 1+(0.9730.230i)T 1 + (-0.973 - 0.230i)T
show more
show less
   L(s)=p (1αpps)1L(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 ZZ-function along the critical line