Properties

Label 1-2664-2664.1757-r0-0-0
Degree 11
Conductor 26642664
Sign 0.5800.814i0.580 - 0.814i
Analytic cond. 12.371512.3715
Root an. cond. 12.371512.3715
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  + (0.342 + 0.939i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)11-s + (0.984 + 0.173i)13-s + (0.342 − 0.939i)17-s + (0.984 + 0.173i)19-s + (0.866 − 0.5i)23-s + (−0.766 + 0.642i)25-s + (−0.866 − 0.5i)29-s + (−0.866 − 0.5i)31-s + (0.984 − 0.173i)35-s + (0.173 − 0.984i)41-s + (0.866 − 0.5i)43-s − 47-s + (−0.939 − 0.342i)49-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)11-s + (0.984 + 0.173i)13-s + (0.342 − 0.939i)17-s + (0.984 + 0.173i)19-s + (0.866 − 0.5i)23-s + (−0.766 + 0.642i)25-s + (−0.866 − 0.5i)29-s + (−0.866 − 0.5i)31-s + (0.984 − 0.173i)35-s + (0.173 − 0.984i)41-s + (0.866 − 0.5i)43-s − 47-s + (−0.939 − 0.342i)49-s + ⋯

Functional equation

Λ(s)=(2664s/2ΓR(s)L(s)=((0.5800.814i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2664s/2ΓR(s)L(s)=((0.5800.814i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 26642664    =    2332372^{3} \cdot 3^{2} \cdot 37
Sign: 0.5800.814i0.580 - 0.814i
Analytic conductor: 12.371512.3715
Root analytic conductor: 12.371512.3715
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2664(1757,)\chi_{2664} (1757, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2664, (0: ), 0.5800.814i)(1,\ 2664,\ (0:\ ),\ 0.580 - 0.814i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.7870469230.9209819060i1.787046923 - 0.9209819060i
L(12)L(\frac12) \approx 1.7870469230.9209819060i1.787046923 - 0.9209819060i
L(1)L(1) \approx 1.2667488160.1638616787i1.266748816 - 0.1638616787i
L(1)L(1) \approx 1.2667488160.1638616787i1.266748816 - 0.1638616787i

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
37 1 1
good5 1+(0.342+0.939i)T 1 + (0.342 + 0.939i)T
7 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
11 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
13 1+(0.984+0.173i)T 1 + (0.984 + 0.173i)T
17 1+(0.3420.939i)T 1 + (0.342 - 0.939i)T
19 1+(0.984+0.173i)T 1 + (0.984 + 0.173i)T
23 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
29 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
31 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
41 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
43 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
47 1T 1 - T
53 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
59 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
61 1+(0.6420.766i)T 1 + (0.642 - 0.766i)T
67 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
71 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
73 1T 1 - T
79 1+(0.9840.173i)T 1 + (-0.984 - 0.173i)T
83 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
89 1+(0.6420.766i)T 1 + (-0.642 - 0.766i)T
97 1iT 1 - iT
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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

−19.52509235190191143822643768867, −18.64350408337510424136261304220, −17.835262058971030677601383112942, −17.53105592396844917309568334416, −16.41068284037348856124638653423, −16.10095365609329026129711336975, −14.99979669114173616437490744887, −14.73099834735813099687370748934, −13.5401869511208129511844953047, −12.95255271862714015977815965166, −12.35928662663850941221966292312, −11.65356751286269128889043840505, −10.89468503145473923506637744420, −9.81835181811317629387615829928, −9.20518341851779447448152958118, −8.70533109146204004548694327002, −7.888757037633846076729405340197, −6.98606431753546536310906392658, −5.89158634056349430563134364341, −5.51270290274360306812294113970, −4.66684344678875691115167897648, −3.77457039121442837459546136306, −2.83514117553307943837729541150, −1.58949991617223102070881323201, −1.33464908362865461855691718704, 0.67173549718373974998255348774, 1.55889409917435191359413257201, 2.71279075005694890055165351182, 3.546481689796046377413064652649, 4.00855360358497443743768083010, 5.339290017495637292301806414004, 5.92843775746629074899377030610, 6.94817190555672716002224360662, 7.25887799296095361383359624348, 8.25051010051955787760978482709, 9.18402345040282907471496268507, 9.851417186507439671800710838170, 10.71902105452544246966028277813, 11.245126587514847676374840599571, 11.727263947105716431958996847135, 13.14091709903948739766057623078, 13.545550300108075140087242480455, 14.309199248583292654644590271118, 14.63237758934733041045029111749, 15.841076919448110921102940799532, 16.366660427100981182624792091617, 17.11756306317553354769745073832, 17.83248355531007965717114770039, 18.65036551958964657639245513354, 18.94928257615967752004089157037

Graph of the ZZ-function along the critical line