↑

Properties

Label 2-1560-5.4-c1-0-9
Degree 22
Conductor 15601560
Sign 0.621−0.783i0.621 - 0.783i
Analytic cond. 12.456612.4566
Root an. cond. 3.529393.52939
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  + i·3-s + (−1.75 − 1.38i)5-s − 9-s + 1.50·11-s + i·13-s + (1.38 − 1.75i)15-s + 2.72i·17-s − 0.726·19-s − 4.72i·23-s + (1.14 + 4.86i)25-s − i·27-s + 7.55·29-s − 3.00·31-s + 1.50i·33-s + 5.00i·37-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.783 − 0.621i)5-s − 0.333·9-s + 0.453·11-s + 0.277i·13-s + (0.358 − 0.452i)15-s + 0.661i·17-s − 0.166·19-s − 0.985i·23-s + (0.228 + 0.973i)25-s − 0.192i·27-s + 1.40·29-s − 0.540·31-s + 0.261i·33-s + 0.823i·37-s + ⋯

Functional equation

Λ(s)=(1560s/2 ΓC(s) L(s)=((0.621−0.783i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1560s/2 ΓC(s+1/2) L(s)=((0.621−0.783i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15601560    =    23⋅3⋅5⋅132^{3} \cdot 3 \cdot 5 \cdot 13
Sign: 0.621−0.783i0.621 - 0.783i
Analytic conductor: 12.456612.4566
Root analytic conductor: 3.529393.52939
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1560(1249,⋅)\chi_{1560} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1560, ( :1/2), 0.621−0.783i)(2,\ 1560,\ (\ :1/2),\ 0.621 - 0.783i)

Particular Values

L(1)L(1) ≈\approx 1.3414945451.341494545
L(12)L(\frac12) ≈\approx 1.3414945451.341494545
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
3 1−iT 1 - iT
5 1+(1.75+1.38i)T 1 + (1.75 + 1.38i)T
13 1−iT 1 - iT
good7 1−7T2 1 - 7T^{2}
11 1−1.50T+11T2 1 - 1.50T + 11T^{2}
17 1−2.72iT−17T2 1 - 2.72iT - 17T^{2}
19 1+0.726T+19T2 1 + 0.726T + 19T^{2}
23 1+4.72iT−23T2 1 + 4.72iT - 23T^{2}
29 1−7.55T+29T2 1 - 7.55T + 29T^{2}
31 1+3.00T+31T2 1 + 3.00T + 31T^{2}
37 1−5.00iT−37T2 1 - 5.00iT - 37T^{2}
41 1−5.78T+41T2 1 - 5.78T + 41T^{2}
43 1−2.72iT−43T2 1 - 2.72iT - 43T^{2}
47 1−10.2iT−47T2 1 - 10.2iT - 47T^{2}
53 1+7.55iT−53T2 1 + 7.55iT - 53T^{2}
59 1−12.5T+59T2 1 - 12.5T + 59T^{2}
61 1−6.28T+61T2 1 - 6.28T + 61T^{2}
67 1−12.5iT−67T2 1 - 12.5iT - 67T^{2}
71 1−4.77T+71T2 1 - 4.77T + 71T^{2}
73 1−12.0iT−73T2 1 - 12.0iT - 73T^{2}
79 1+5.27T+79T2 1 + 5.27T + 79T^{2}
83 1−7.78iT−83T2 1 - 7.78iT - 83T^{2}
89 1+1.78T+89T2 1 + 1.78T + 89T^{2}
97 1−6iT−97T2 1 - 6iT - 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

−9.529483734734021801915179388985, −8.555382327827838305792906697918, −8.330674430023863386139069887098, −7.15077712551556995085925016431, −6.30550584023024407782859623276, −5.26696690886148509486899115733, −4.36223527195295732273681589980, −3.87403364941065329325446591857, −2.62788764105150553599212179629, −1.02576401867867564325747385002, 0.67100685482534905698289186302, 2.22509669275629148811609855287, 3.25242509453202029788493913655, 4.10730708062395728568942631491, 5.26525440344810809138777868042, 6.23384760064478230719889467819, 7.08525839282254982372441470937, 7.54117328949581074182351928452, 8.436693432652361115141152991773, 9.192065201062430856056753813061

Graph of the ZZ-function along the critical line