Properties

Label 1-693-693.38-r0-0-0
Degree 11
Conductor 693693
Sign 0.999+0.0111i0.999 + 0.0111i
Analytic cond. 3.218273.21827
Root an. cond. 3.218273.21827
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)10-s + (0.104 + 0.994i)13-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.669 + 0.743i)26-s + (0.978 − 0.207i)29-s + (0.809 − 0.587i)31-s − 32-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + (−0.309 − 0.951i)8-s + (0.5 + 0.866i)10-s + (0.104 + 0.994i)13-s + (−0.809 − 0.587i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (0.913 + 0.406i)20-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.669 + 0.743i)26-s + (0.978 − 0.207i)29-s + (0.809 − 0.587i)31-s − 32-s + ⋯

Functional equation

Λ(s)=(693s/2ΓR(s)L(s)=((0.999+0.0111i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0111i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(693s/2ΓR(s)L(s)=((0.999+0.0111i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0111i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 693693    =    327113^{2} \cdot 7 \cdot 11
Sign: 0.999+0.0111i0.999 + 0.0111i
Analytic conductor: 3.218273.21827
Root analytic conductor: 3.218273.21827
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ693(38,)\chi_{693} (38, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 693, (0: ), 0.999+0.0111i)(1,\ 693,\ (0:\ ),\ 0.999 + 0.0111i)

Particular Values

L(12)L(\frac{1}{2}) \approx 2.213881660+0.01237308261i2.213881660 + 0.01237308261i
L(12)L(\frac12) \approx 2.213881660+0.01237308261i2.213881660 + 0.01237308261i
L(1)L(1) \approx 1.6183733190.2038402562i1.618373319 - 0.2038402562i
L(1)L(1) \approx 1.6183733190.2038402562i1.618373319 - 0.2038402562i

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
11 1 1
good2 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
5 1+(0.104+0.994i)T 1 + (-0.104 + 0.994i)T
13 1+(0.104+0.994i)T 1 + (0.104 + 0.994i)T
17 1+(0.104+0.994i)T 1 + (-0.104 + 0.994i)T
19 1+(0.978+0.207i)T 1 + (0.978 + 0.207i)T
23 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
29 1+(0.9780.207i)T 1 + (0.978 - 0.207i)T
31 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
37 1+(0.669+0.743i)T 1 + (0.669 + 0.743i)T
41 1+(0.9780.207i)T 1 + (-0.978 - 0.207i)T
43 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
47 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
53 1+(0.9130.406i)T 1 + (-0.913 - 0.406i)T
59 1+(0.3090.951i)T 1 + (0.309 - 0.951i)T
61 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
67 1+T 1 + T
71 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
73 1+(0.9780.207i)T 1 + (0.978 - 0.207i)T
79 1+(0.809+0.587i)T 1 + (-0.809 + 0.587i)T
83 1+(0.104+0.994i)T 1 + (-0.104 + 0.994i)T
89 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
97 1+(0.9130.406i)T 1 + (-0.913 - 0.406i)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

−22.908857514787956245096668635411, −21.942015187597164814853685290661, −21.10484546932752974678234301799, −20.31755155391102220399966061829, −19.87573486247376755282297904132, −18.32589451253294603362136137441, −17.587315844773553370765030557957, −16.71185886426569663524082414656, −15.966349298681529609133972757072, −15.448194954698315154074779993047, −14.3259079919386297692609343951, −13.54105014842531231162724828352, −12.80844814299202206428276372516, −12.08394628937959625017773344564, −11.28516253957277109146090261520, −9.97471066384843855063814932736, −8.84358689083618125626402938734, −8.145950259343070266876452602468, −7.25424955174794559743089202334, −6.22766763084551298579239174760, −5.10722852761659233975230206421, −4.78736901260554166463130038158, −3.48447728329455576560395242052, −2.57781176463634300448845659211, −0.90542689687867168897558352780, 1.373332346507290039085328353473, 2.43570526421544571971300286684, 3.39423063612475982125602097569, 4.16114376899510061188800282013, 5.30708956733430886676836588758, 6.35678011852847890792407422291, 6.92953896198014103717194473571, 8.16360835011232837878126546767, 9.58280423426148386923772287512, 10.162022126837895394322211623265, 11.263225296393655920906254398737, 11.62384992652217282254950285417, 12.679512876261690756247570033928, 13.786538874296533483579858781712, 14.13346127876365704661949371374, 15.21348941595111058874368549963, 15.670751439703252286965810495437, 16.93885217723983471926630639389, 18.016899945518633732483544549205, 18.96052249160934014101249630173, 19.30031222209300461458770950223, 20.34485206112707674288525755824, 21.24496983580773063803100513419, 21.92339965216829295883727346650, 22.49129574233981375857861659314

Graph of the ZZ-function along the critical line