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Properties

Label 1-6145-6145.1029-r1-0-0
Degree 11
Conductor 61456145
Sign 0.974−0.225i0.974 - 0.225i
Analytic cond. 660.371660.371
Root an. cond. 660.371660.371
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.302 + 0.953i)2-s + (0.233 + 0.972i)3-s + (−0.817 − 0.576i)4-s + (−0.997 − 0.0715i)6-s + (0.555 + 0.831i)7-s + (0.796 − 0.605i)8-s + (−0.891 + 0.453i)9-s + (0.641 − 0.767i)11-s + (0.369 − 0.929i)12-s + (0.272 + 0.962i)13-s + (−0.960 + 0.277i)14-s + (0.336 + 0.941i)16-s + (−0.902 − 0.430i)17-s + (−0.163 − 0.986i)18-s + (−0.853 + 0.520i)19-s + ⋯
L(s)  = 1  + (−0.302 + 0.953i)2-s + (0.233 + 0.972i)3-s + (−0.817 − 0.576i)4-s + (−0.997 − 0.0715i)6-s + (0.555 + 0.831i)7-s + (0.796 − 0.605i)8-s + (−0.891 + 0.453i)9-s + (0.641 − 0.767i)11-s + (0.369 − 0.929i)12-s + (0.272 + 0.962i)13-s + (−0.960 + 0.277i)14-s + (0.336 + 0.941i)16-s + (−0.902 − 0.430i)17-s + (−0.163 − 0.986i)18-s + (−0.853 + 0.520i)19-s + ⋯

Functional equation

Λ(s)=(6145s/2 ΓR(s+1) L(s)=((0.974−0.225i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(6145s/2 ΓR(s+1) L(s)=((0.974−0.225i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 61456145    =    5⋅12295 \cdot 1229
Sign: 0.974−0.225i0.974 - 0.225i
Analytic conductor: 660.371660.371
Root analytic conductor: 660.371660.371
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ6145(1029,⋅)\chi_{6145} (1029, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 6145, (1: ), 0.974−0.225i)(1,\ 6145,\ (1:\ ),\ 0.974 - 0.225i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx −0.07685547855+0.008759032058i-0.07685547855 + 0.008759032058i
L(12)L(\frac12) ≈\approx −0.07685547855+0.008759032058i-0.07685547855 + 0.008759032058i
L(1)L(1) ≈\approx 0.5303466082+0.6805377874i0.5303466082 + 0.6805377874i
L(1)L(1) ≈\approx 0.5303466082+0.6805377874i0.5303466082 + 0.6805377874i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
1229 1 1
good2 1+(−0.302+0.953i)T 1 + (-0.302 + 0.953i)T
3 1+(0.233+0.972i)T 1 + (0.233 + 0.972i)T
7 1+(0.555+0.831i)T 1 + (0.555 + 0.831i)T
11 1+(0.641−0.767i)T 1 + (0.641 - 0.767i)T
13 1+(0.272+0.962i)T 1 + (0.272 + 0.962i)T
17 1+(−0.902−0.430i)T 1 + (-0.902 - 0.430i)T
19 1+(−0.853+0.520i)T 1 + (-0.853 + 0.520i)T
23 1+(−0.122+0.992i)T 1 + (-0.122 + 0.992i)T
29 1+(0.350+0.936i)T 1 + (0.350 + 0.936i)T
31 1+(0.900+0.435i)T 1 + (0.900 + 0.435i)T
37 1+(0.444−0.895i)T 1 + (0.444 - 0.895i)T
41 1+(0.823+0.567i)T 1 + (0.823 + 0.567i)T
43 1+(−0.981−0.193i)T 1 + (-0.981 - 0.193i)T
47 1+(0.580+0.814i)T 1 + (0.580 + 0.814i)T
53 1+(−0.881+0.471i)T 1 + (-0.881 + 0.471i)T
59 1+(0.439−0.898i)T 1 + (0.439 - 0.898i)T
61 1+(0.917−0.397i)T 1 + (0.917 - 0.397i)T
67 1+(−0.959−0.282i)T 1 + (-0.959 - 0.282i)T
71 1+(0.848+0.529i)T 1 + (0.848 + 0.529i)T
73 1+(−0.0664−0.997i)T 1 + (-0.0664 - 0.997i)T
79 1+(0.605−0.796i)T 1 + (0.605 - 0.796i)T
83 1+(−0.843−0.537i)T 1 + (-0.843 - 0.537i)T
89 1+(−0.747−0.664i)T 1 + (-0.747 - 0.664i)T
97 1+(0.238−0.971i)T 1 + (0.238 - 0.971i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−17.24949881706744868410129868651, −16.690576070872542612034132232349, −15.304670289560582183044250814311, −14.79796313169997953006693389149, −13.99667446042758103167116648100, −13.383636935368486122378827065369, −12.96160223317656453497818058896, −12.28949457108366389032258196354, −11.59311640415623004184288365559, −11.02763682134180214821261015681, −10.3280401516504909545757121442, −9.704915139435293415171896508700, −8.64008672220877947723315912234, −8.34891068738985783394579902167, −7.685099372102384846545731819661, −6.84676966414148843286404221789, −6.308017954770030542802233212701, −5.11593144126441569384783349225, −4.270062608186546601287898161298, −3.8759787248491851536527323377, −2.64349321619318647163068451361, −2.29145298654094932124638911302, −1.35377084090596449709773621936, −0.77478358291687371495121971670, −0.01352558836470393194990753126, 1.28921048472071760892649737740, 2.11744194923966656640331605128, 3.19488088662040300730403613634, 4.064132783829177994885763149816, 4.607907118038374971129785361167, 5.270817442394909689302168027552, 6.10450140913252243884135054923, 6.473920506993552106218906968, 7.580197569941270955507653152646, 8.37207974007129023940436049059, 8.84027479633183624552740861958, 9.2292105834370966201000860555, 9.92987806781647303855274966926, 10.96647874757450675518586943561, 11.243119684992227327691136686, 12.13398009697003623670379131712, 13.17398898448520776737000981326, 14.04105829121236718467325252512, 14.30113942950993788399075806840, 14.924159491760409581619230181764, 15.76040769260651825107245752772, 15.97543764795583231578033061908, 16.720330096334236576558116357874, 17.35529111184070555388321659516, 17.940149426645376881753656352856

Graph of the ZZ-function along the critical line