Properties

Label 1-6384-6384.4421-r0-0-0
Degree 11
Conductor 63846384
Sign 0.5560.830i0.556 - 0.830i
Analytic cond. 29.647129.6471
Root an. cond. 29.647129.6471
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  + (−0.984 − 0.173i)5-s i·11-s + (0.642 + 0.766i)13-s + (0.939 − 0.342i)17-s + (0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (0.984 − 0.173i)29-s + (0.5 − 0.866i)31-s + (0.866 + 0.5i)37-s + (−0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.939 + 0.342i)47-s + (−0.984 + 0.173i)53-s + (−0.173 + 0.984i)55-s + (−0.342 − 0.939i)59-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)5-s i·11-s + (0.642 + 0.766i)13-s + (0.939 − 0.342i)17-s + (0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (0.984 − 0.173i)29-s + (0.5 − 0.866i)31-s + (0.866 + 0.5i)37-s + (−0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.939 + 0.342i)47-s + (−0.984 + 0.173i)53-s + (−0.173 + 0.984i)55-s + (−0.342 − 0.939i)59-s + ⋯

Functional equation

Λ(s)=(6384s/2ΓR(s)L(s)=((0.5560.830i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(6384s/2ΓR(s)L(s)=((0.5560.830i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 63846384    =    2437192^{4} \cdot 3 \cdot 7 \cdot 19
Sign: 0.5560.830i0.556 - 0.830i
Analytic conductor: 29.647129.6471
Root analytic conductor: 29.647129.6471
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ6384(4421,)\chi_{6384} (4421, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 6384, (0: ), 0.5560.830i)(1,\ 6384,\ (0:\ ),\ 0.556 - 0.830i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.4621403250.7801942553i1.462140325 - 0.7801942553i
L(12)L(\frac12) \approx 1.4621403250.7801942553i1.462140325 - 0.7801942553i
L(1)L(1) \approx 1.0117452940.1572129498i1.011745294 - 0.1572129498i
L(1)L(1) \approx 1.0117452940.1572129498i1.011745294 - 0.1572129498i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
7 1 1
19 1 1
good5 1+(0.9840.173i)T 1 + (-0.984 - 0.173i)T
11 1iT 1 - iT
13 1+(0.642+0.766i)T 1 + (0.642 + 0.766i)T
17 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
23 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
29 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
31 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
37 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
41 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
43 1+(0.3420.939i)T 1 + (-0.342 - 0.939i)T
47 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
53 1+(0.984+0.173i)T 1 + (-0.984 + 0.173i)T
59 1+(0.3420.939i)T 1 + (-0.342 - 0.939i)T
61 1+(0.6420.766i)T 1 + (-0.642 - 0.766i)T
67 1+(0.642+0.766i)T 1 + (0.642 + 0.766i)T
71 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
73 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
79 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
83 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
89 1+(0.1730.984i)T 1 + (-0.173 - 0.984i)T
97 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)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

−17.96140153687661869688119054474, −16.955383284094044982069384087712, −16.40062994588271793492169805366, −15.56567083893767664602487821973, −15.223250593953587323089596381714, −14.63333375990882846405403739490, −13.82468582474277472187146006527, −13.01452451719799532067874451742, −12.346297795977560422131705534676, −11.97904357417837522002414899505, −11.048598565992540297837217153274, −10.57516173504817047134138137880, −9.8432507858685457408731402261, −9.0836807300146914357765289536, −8.11344995863137608255281731845, −7.9000453090036320947263596554, −7.0270665076616598085673522208, −6.44214451413306837119863396153, −5.46900134535313342936864162041, −4.78287883082557324238935767654, −4.08956345004920239390214188114, −3.24682962315173024236838324818, −2.826585689632739196993424841858, −1.532243809562216880143864621019, −0.863213666397516738694208682074, 0.59679203075880184256121467807, 1.16870014498054614407652339327, 2.41443861358874561196418702789, 3.2487248911620359715941926339, 3.76931000490613298335505762780, 4.585555413096241797879118345771, 5.22614980047788836530544382319, 6.21274200209819930160010598474, 6.70268878758377887555522908673, 7.64377228794236905110090218265, 8.19189519870745932976850355270, 8.7725163375532700152927893108, 9.44039452031512024745282514013, 10.39233791663695490564622722856, 11.05513898145465892856570136906, 11.59962779425397344658169993618, 12.15002913078597703230649014280, 12.864443138972833446100829752793, 13.743814278629911730219720468704, 14.13046746063773618840827816966, 14.99059279228035770081729568066, 15.67089624493083902694064054372, 16.16109255758950030198472685933, 16.80125144264204281318316634373, 17.25244492102289413482509261569

Graph of the ZZ-function along the critical line