Properties

Label 1-1312-1312.379-r1-0-0
Degree 11
Conductor 13121312
Sign 0.6430.765i0.643 - 0.765i
Analytic cond. 140.993140.993
Root an. cond. 140.993140.993
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  + (0.707 − 0.707i)3-s + (−0.987 − 0.156i)5-s + (−0.951 − 0.309i)7-s i·9-s + (−0.156 − 0.987i)11-s + (0.891 + 0.453i)13-s + (−0.809 + 0.587i)15-s + (0.809 + 0.587i)17-s + (0.453 + 0.891i)19-s + (−0.891 + 0.453i)21-s + (−0.951 + 0.309i)23-s + (0.951 + 0.309i)25-s + (−0.707 − 0.707i)27-s + (0.156 − 0.987i)29-s + (0.809 + 0.587i)31-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.987 − 0.156i)5-s + (−0.951 − 0.309i)7-s i·9-s + (−0.156 − 0.987i)11-s + (0.891 + 0.453i)13-s + (−0.809 + 0.587i)15-s + (0.809 + 0.587i)17-s + (0.453 + 0.891i)19-s + (−0.891 + 0.453i)21-s + (−0.951 + 0.309i)23-s + (0.951 + 0.309i)25-s + (−0.707 − 0.707i)27-s + (0.156 − 0.987i)29-s + (0.809 + 0.587i)31-s + ⋯

Functional equation

Λ(s)=(1312s/2ΓR(s+1)L(s)=((0.6430.765i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1312s/2ΓR(s+1)L(s)=((0.6430.765i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 13121312    =    25412^{5} \cdot 41
Sign: 0.6430.765i0.643 - 0.765i
Analytic conductor: 140.993140.993
Root analytic conductor: 140.993140.993
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1312(379,)\chi_{1312} (379, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1312, (1: ), 0.6430.765i)(1,\ 1312,\ (1:\ ),\ 0.643 - 0.765i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.7599449160.8193753540i1.759944916 - 0.8193753540i
L(12)L(\frac12) \approx 1.7599449160.8193753540i1.759944916 - 0.8193753540i
L(1)L(1) \approx 1.0353431690.3256899177i1.035343169 - 0.3256899177i
L(1)L(1) \approx 1.0353431690.3256899177i1.035343169 - 0.3256899177i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
41 1 1
good3 1+(0.7070.707i)T 1 + (0.707 - 0.707i)T
5 1+(0.9870.156i)T 1 + (-0.987 - 0.156i)T
7 1+(0.9510.309i)T 1 + (-0.951 - 0.309i)T
11 1+(0.1560.987i)T 1 + (-0.156 - 0.987i)T
13 1+(0.891+0.453i)T 1 + (0.891 + 0.453i)T
17 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
19 1+(0.453+0.891i)T 1 + (0.453 + 0.891i)T
23 1+(0.951+0.309i)T 1 + (-0.951 + 0.309i)T
29 1+(0.1560.987i)T 1 + (0.156 - 0.987i)T
31 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
37 1+(0.9870.156i)T 1 + (-0.987 - 0.156i)T
43 1+(0.453+0.891i)T 1 + (-0.453 + 0.891i)T
47 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
53 1+(0.987+0.156i)T 1 + (0.987 + 0.156i)T
59 1+(0.4530.891i)T 1 + (0.453 - 0.891i)T
61 1+(0.453+0.891i)T 1 + (0.453 + 0.891i)T
67 1+(0.156+0.987i)T 1 + (-0.156 + 0.987i)T
71 1+(0.587+0.809i)T 1 + (-0.587 + 0.809i)T
73 1iT 1 - iT
79 1+T 1 + T
83 1+(0.707+0.707i)T 1 + (-0.707 + 0.707i)T
89 1+(0.951+0.309i)T 1 + (0.951 + 0.309i)T
97 1+(0.809+0.587i)T 1 + (-0.809 + 0.587i)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

−20.57727571899359415969857841704, −20.19072011090483413618814172192, −19.52795491249843516354682442436, −18.70946574110246904765219026579, −18.08325007056855250200703561870, −16.7458009900828510493278034359, −16.01307321379976746825372990020, −15.54651862415098967986104458447, −15.03440373613298920641307539965, −14.00929667180078972709905318716, −13.259603745834765070002019072992, −12.316292613788573925429232591980, −11.65007077622477392263330115006, −10.47562756458091917794494498041, −10.036430405021205510995052030767, −9.05685661321462814959519188853, −8.39763364301398902655929098823, −7.48837772155773139875829038327, −6.797632462891758635528985294758, −5.495029787373209223819351066805, −4.643117707057122294109340130735, −3.65237360252684774580333283968, −3.17486574958548957986669286250, −2.21449156199078165977557412603, −0.58178930859895088389242889904, 0.63108227002479418980613089161, 1.42164170166692332929495320639, 2.880567886585944853454196696339, 3.5929070922495526910819814953, 4.045392713113057121791793446914, 5.76257575141177459552421901651, 6.38203242397527403404036806621, 7.332251534378993314498526050785, 8.15053745619574384441411864634, 8.555296578818138586732380316153, 9.63201977063030157975412801860, 10.4803203796851856858706743596, 11.64066077928876238881271918216, 12.160771126500987159425676010861, 13.01101602004431467824078566606, 13.750880185827133246172496631827, 14.31520623011933655082048595112, 15.42115305314137764109043105616, 16.08588593880786344212767543218, 16.60363012088772367850855736691, 17.80297543414161399650240448421, 18.79878837067898744935162018689, 19.15690951229053032829056043342, 19.626656085361434807942969137006, 20.63135646781700842141958351149

Graph of the ZZ-function along the critical line