↑

Properties

Label 2-1408-176.43-c1-0-40
Degree 22
Conductor 14081408
Sign −0.981+0.192i-0.981 + 0.192i
Analytic cond. 11.242911.2429
Root an. cond. 3.353043.35304
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.15 − 1.15i)3-s + (2.51 + 2.51i)5-s − 4.37i·7-s − 0.313i·9-s + (−3.31 − 0.130i)11-s + (−2.19 − 2.19i)13-s − 5.82i·15-s + 0.470i·17-s + (−0.632 + 0.632i)19-s + (−5.07 + 5.07i)21-s − 6.62·23-s + 7.61i·25-s + (−3.84 + 3.84i)27-s + (−0.0549 − 0.0549i)29-s − 0.184i·31-s + ⋯
L(s)  = 1  + (−0.669 − 0.669i)3-s + (1.12 + 1.12i)5-s − 1.65i·7-s − 0.104i·9-s + (−0.999 − 0.0393i)11-s + (−0.609 − 0.609i)13-s − 1.50i·15-s + 0.114i·17-s + (−0.145 + 0.145i)19-s + (−1.10 + 1.10i)21-s − 1.38·23-s + 1.52i·25-s + (−0.739 + 0.739i)27-s + (−0.0102 − 0.0102i)29-s − 0.0331i·31-s + ⋯

Functional equation

Λ(s)=(1408s/2 ΓC(s) L(s)=((−0.981+0.192i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1408s/2 ΓC(s+1/2) L(s)=((−0.981+0.192i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14081408    =    27⋅112^{7} \cdot 11
Sign: −0.981+0.192i-0.981 + 0.192i
Analytic conductor: 11.242911.2429
Root analytic conductor: 3.353043.35304
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1408(1055,⋅)\chi_{1408} (1055, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1408, ( :1/2), −0.981+0.192i)(2,\ 1408,\ (\ :1/2),\ -0.981 + 0.192i)

Particular Values

L(1)L(1) ≈\approx 0.65043774700.6504377470
L(12)L(\frac12) ≈\approx 0.65043774700.6504377470
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
11 1+(3.31+0.130i)T 1 + (3.31 + 0.130i)T
good3 1+(1.15+1.15i)T+3iT2 1 + (1.15 + 1.15i)T + 3iT^{2}
5 1+(−2.51−2.51i)T+5iT2 1 + (-2.51 - 2.51i)T + 5iT^{2}
7 1+4.37iT−7T2 1 + 4.37iT - 7T^{2}
13 1+(2.19+2.19i)T+13iT2 1 + (2.19 + 2.19i)T + 13iT^{2}
17 1−0.470iT−17T2 1 - 0.470iT - 17T^{2}
19 1+(0.632−0.632i)T−19iT2 1 + (0.632 - 0.632i)T - 19iT^{2}
23 1+6.62T+23T2 1 + 6.62T + 23T^{2}
29 1+(0.0549+0.0549i)T+29iT2 1 + (0.0549 + 0.0549i)T + 29iT^{2}
31 1+0.184iT−31T2 1 + 0.184iT - 31T^{2}
37 1+(3.55+3.55i)T+37iT2 1 + (3.55 + 3.55i)T + 37iT^{2}
41 1−3.77T+41T2 1 - 3.77T + 41T^{2}
43 1+(−4.82−4.82i)T+43iT2 1 + (-4.82 - 4.82i)T + 43iT^{2}
47 1−0.258iT−47T2 1 - 0.258iT - 47T^{2}
53 1+(5.27+5.27i)T+53iT2 1 + (5.27 + 5.27i)T + 53iT^{2}
59 1+(−1.64+1.64i)T−59iT2 1 + (-1.64 + 1.64i)T - 59iT^{2}
61 1+(9.05+9.05i)T+61iT2 1 + (9.05 + 9.05i)T + 61iT^{2}
67 1+(1.83+1.83i)T+67iT2 1 + (1.83 + 1.83i)T + 67iT^{2}
71 1−13.1T+71T2 1 - 13.1T + 71T^{2}
73 1+10.7T+73T2 1 + 10.7T + 73T^{2}
79 1−1.80T+79T2 1 - 1.80T + 79T^{2}
83 1+(8.61−8.61i)T−83iT2 1 + (8.61 - 8.61i)T - 83iT^{2}
89 1−14.8iT−89T2 1 - 14.8iT - 89T^{2}
97 1+9.13T+97T2 1 + 9.13T + 97T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.618792980739799168997875159664, −7.945506782845948305033440101852, −7.43105546533549836621949054346, −6.67642863378061894463314130257, −6.07242112049273884135203150490, −5.25021705846912243340867949209, −3.94114959236865649260506400877, −2.85130920901725258489155441373, −1.71108756537989230322167093228, −0.26168129366855675831137924855, 1.90163899193552335997898757252, 2.56959482512589604338719243421, 4.44354075863664054929190780305, 5.06919143258309035331213719388, 5.67496477715914504351444710782, 6.12858487244138033752132547040, 7.67325371245069764265661364294, 8.632368934720684850015941003524, 9.182379869321908811580539227701, 9.900156767130795178484296974233

Graph of the ZZ-function along the critical line