Properties

Label 1-2793-2793.452-r0-0-0
Degree 11
Conductor 27932793
Sign 0.9630.268i-0.963 - 0.268i
Analytic cond. 12.970612.9706
Root an. cond. 12.970612.9706
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.698 − 0.715i)2-s + (−0.0249 − 0.999i)4-s + (0.797 − 0.603i)5-s + (−0.733 − 0.680i)8-s + (0.124 − 0.992i)10-s + (0.900 − 0.433i)11-s + (−0.995 + 0.0995i)13-s + (−0.998 + 0.0498i)16-s + (−0.878 + 0.478i)17-s + (−0.623 − 0.781i)20-s + (0.318 − 0.947i)22-s + (0.853 − 0.521i)23-s + (0.270 − 0.962i)25-s + (−0.623 + 0.781i)26-s + (0.980 + 0.198i)29-s + ⋯
L(s)  = 1  + (0.698 − 0.715i)2-s + (−0.0249 − 0.999i)4-s + (0.797 − 0.603i)5-s + (−0.733 − 0.680i)8-s + (0.124 − 0.992i)10-s + (0.900 − 0.433i)11-s + (−0.995 + 0.0995i)13-s + (−0.998 + 0.0498i)16-s + (−0.878 + 0.478i)17-s + (−0.623 − 0.781i)20-s + (0.318 − 0.947i)22-s + (0.853 − 0.521i)23-s + (0.270 − 0.962i)25-s + (−0.623 + 0.781i)26-s + (0.980 + 0.198i)29-s + ⋯

Functional equation

Λ(s)=(2793s/2ΓR(s)L(s)=((0.9630.268i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2ΓR(s)L(s)=((0.9630.268i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 27932793    =    372193 \cdot 7^{2} \cdot 19
Sign: 0.9630.268i-0.963 - 0.268i
Analytic conductor: 12.970612.9706
Root analytic conductor: 12.970612.9706
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(452,)\chi_{2793} (452, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2793, (0: ), 0.9630.268i)(1,\ 2793,\ (0:\ ),\ -0.963 - 0.268i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.34236855962.507460673i0.3423685596 - 2.507460673i
L(12)L(\frac12) \approx 0.34236855962.507460673i0.3423685596 - 2.507460673i
L(1)L(1) \approx 1.2151739071.094650517i1.215173907 - 1.094650517i
L(1)L(1) \approx 1.2151739071.094650517i1.215173907 - 1.094650517i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
7 1 1
19 1 1
good2 1+(0.6980.715i)T 1 + (0.698 - 0.715i)T
5 1+(0.7970.603i)T 1 + (0.797 - 0.603i)T
11 1+(0.9000.433i)T 1 + (0.900 - 0.433i)T
13 1+(0.995+0.0995i)T 1 + (-0.995 + 0.0995i)T
17 1+(0.878+0.478i)T 1 + (-0.878 + 0.478i)T
23 1+(0.8530.521i)T 1 + (0.853 - 0.521i)T
29 1+(0.980+0.198i)T 1 + (0.980 + 0.198i)T
31 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
37 1+(0.9880.149i)T 1 + (0.988 - 0.149i)T
41 1+(0.1240.992i)T 1 + (-0.124 - 0.992i)T
43 1+(0.998+0.0498i)T 1 + (-0.998 + 0.0498i)T
47 1+(0.583+0.811i)T 1 + (0.583 + 0.811i)T
53 1+(0.02490.999i)T 1 + (-0.0249 - 0.999i)T
59 1+(0.998+0.0498i)T 1 + (-0.998 + 0.0498i)T
61 1+(0.3180.947i)T 1 + (-0.318 - 0.947i)T
67 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
71 1+(0.6610.749i)T 1 + (-0.661 - 0.749i)T
73 1+(0.4110.911i)T 1 + (-0.411 - 0.911i)T
79 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
83 1+(0.07470.997i)T 1 + (-0.0747 - 0.997i)T
89 1+(0.969+0.246i)T 1 + (-0.969 + 0.246i)T
97 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
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   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

−19.735827393654214572419565211, −18.61050984870820045641342701451, −17.87859100813179268408081896721, −17.27985926286564099733643985761, −16.89672100458898339143453975931, −15.88979064847963104946308990716, −15.02608472968311629440611797477, −14.74858317735971788472979820090, −13.853234377706893514101876561454, −13.45532584660062766459776150741, −12.55050261495307521038073822016, −11.84476352443279635747990437955, −11.153496422324754153904371987790, −10.099458827952214433270744122735, −9.39852877489081373881132498702, −8.71823244035882049415396885465, −7.65913798240669212084547990132, −6.80427477468414847672853528617, −6.6385708535091830802985881687, −5.60040935511478559692554394771, −4.842309309475478059759013234048, −4.20380157573138844712953659220, −2.98492317898736441975778003605, −2.58610282523337214339472314747, −1.42213607539465770380445362680, 0.58255376342132951719543504240, 1.527710755414235855612016201184, 2.30654459060206674734208098728, 3.057049789185571167016172445853, 4.24239255985213798262716513994, 4.666658976883466754778209950003, 5.52872305694695974910401648114, 6.3385156053187099230313291521, 6.83450584901840886699138668288, 8.25121053536570841778621810917, 9.09520654766890430277189426548, 9.549440982575412863967935540360, 10.371363268775478093265403459372, 11.09424178900588987477487147180, 11.920715921514961606786474602, 12.521953658202837870737824015136, 13.18513073511158327593990651529, 13.83820493489101401130417029773, 14.48348197505905280781282280705, 15.11278208360697537211069022586, 16.0326344355768360707546154465, 16.963770999427403660028049833, 17.37812225314010070491998602024, 18.30682662352397637414954056454, 19.15585540002637464131699708274

Graph of the ZZ-function along the critical line