↑

Properties

Label 2-1408-176.43-c1-0-37
Degree 22
Conductor 14081408
Sign 0.400+0.916i0.400 + 0.916i
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  + (0.766 + 0.766i)3-s + (0.946 + 0.946i)5-s − 0.840i·7-s − 1.82i·9-s + (0.809 − 3.21i)11-s + (−4.53 − 4.53i)13-s + 1.45i·15-s + 0.629i·17-s + (2.53 − 2.53i)19-s + (0.644 − 0.644i)21-s − 3.15·23-s − 3.20i·25-s + (3.69 − 3.69i)27-s + (−3.41 − 3.41i)29-s + 2.10i·31-s + ⋯
L(s)  = 1  + (0.442 + 0.442i)3-s + (0.423 + 0.423i)5-s − 0.317i·7-s − 0.608i·9-s + (0.244 − 0.969i)11-s + (−1.25 − 1.25i)13-s + 0.374i·15-s + 0.152i·17-s + (0.582 − 0.582i)19-s + (0.140 − 0.140i)21-s − 0.657·23-s − 0.641i·25-s + (0.711 − 0.711i)27-s + (−0.633 − 0.633i)29-s + 0.377i·31-s + ⋯

Functional equation

Λ(s)=(1408s/2 ΓC(s) L(s)=((0.400+0.916i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1408s/2 ΓC(s+1/2) L(s)=((0.400+0.916i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14081408    =    27⋅112^{7} \cdot 11
Sign: 0.400+0.916i0.400 + 0.916i
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), 0.400+0.916i)(2,\ 1408,\ (\ :1/2),\ 0.400 + 0.916i)

Particular Values

L(1)L(1) ≈\approx 1.6980242371.698024237
L(12)L(\frac12) ≈\approx 1.6980242371.698024237
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+(−0.809+3.21i)T 1 + (-0.809 + 3.21i)T
good3 1+(−0.766−0.766i)T+3iT2 1 + (-0.766 - 0.766i)T + 3iT^{2}
5 1+(−0.946−0.946i)T+5iT2 1 + (-0.946 - 0.946i)T + 5iT^{2}
7 1+0.840iT−7T2 1 + 0.840iT - 7T^{2}
13 1+(4.53+4.53i)T+13iT2 1 + (4.53 + 4.53i)T + 13iT^{2}
17 1−0.629iT−17T2 1 - 0.629iT - 17T^{2}
19 1+(−2.53+2.53i)T−19iT2 1 + (-2.53 + 2.53i)T - 19iT^{2}
23 1+3.15T+23T2 1 + 3.15T + 23T^{2}
29 1+(3.41+3.41i)T+29iT2 1 + (3.41 + 3.41i)T + 29iT^{2}
31 1−2.10iT−31T2 1 - 2.10iT - 31T^{2}
37 1+(6.49+6.49i)T+37iT2 1 + (6.49 + 6.49i)T + 37iT^{2}
41 1+7.84T+41T2 1 + 7.84T + 41T^{2}
43 1+(−2.55−2.55i)T+43iT2 1 + (-2.55 - 2.55i)T + 43iT^{2}
47 1−3.26iT−47T2 1 - 3.26iT - 47T^{2}
53 1+(−7.27−7.27i)T+53iT2 1 + (-7.27 - 7.27i)T + 53iT^{2}
59 1+(6.06−6.06i)T−59iT2 1 + (6.06 - 6.06i)T - 59iT^{2}
61 1+(−3.33−3.33i)T+61iT2 1 + (-3.33 - 3.33i)T + 61iT^{2}
67 1+(−8.95−8.95i)T+67iT2 1 + (-8.95 - 8.95i)T + 67iT^{2}
71 1−4.32T+71T2 1 - 4.32T + 71T^{2}
73 1+3.78T+73T2 1 + 3.78T + 73T^{2}
79 1−10.7T+79T2 1 - 10.7T + 79T^{2}
83 1+(−9.42+9.42i)T−83iT2 1 + (-9.42 + 9.42i)T - 83iT^{2}
89 1+1.82iT−89T2 1 + 1.82iT - 89T^{2}
97 1+2.21T+97T2 1 + 2.21T + 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

−9.445182449181345144778314439164, −8.731562304724771951930568462973, −7.82101074084809742259106904647, −7.02061345438413187229391786239, −6.04834759537932756857580802003, −5.31245117627083319738260578133, −4.10666568471046997747276746059, −3.24127067101652690315890506494, −2.45355009595946558807638904727, −0.62105182015329204308635363518, 1.78204855930154677053181464814, 2.15327441759664958686422717123, 3.62707753942757530953468850886, 4.88114683763008914523733078020, 5.30391153990423649872407897114, 6.70930445881122969972741058178, 7.24558524689081187759591624309, 8.065136715230401620209957419110, 8.982342605344008813283336678574, 9.615890769856648536599994967932

Graph of the ZZ-function along the critical line