↑

Properties

Label 2-1560-5.4-c1-0-3
Degree 22
Conductor 15601560
Sign −0.981+0.193i-0.981 + 0.193i
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 + (0.432 + 2.19i)5-s − 9-s − 2.86·11-s + i·13-s + (−2.19 + 0.432i)15-s + 5.52i·17-s − 3.52·19-s − 7.52i·23-s + (−4.62 + 1.89i)25-s − i·27-s − 6.77·29-s + 5.72·31-s − 2.86i·33-s − 3.72i·37-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.193 + 0.981i)5-s − 0.333·9-s − 0.863·11-s + 0.277i·13-s + (−0.566 + 0.111i)15-s + 1.33i·17-s − 0.808·19-s − 1.56i·23-s + (−0.925 + 0.379i)25-s − 0.192i·27-s − 1.25·29-s + 1.02·31-s − 0.498i·33-s − 0.613i·37-s + ⋯

Functional equation

Λ(s)=(1560s/2 ΓC(s) L(s)=((−0.981+0.193i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1560s/2 ΓC(s+1/2) L(s)=((−0.981+0.193i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15601560    =    23⋅3⋅5⋅132^{3} \cdot 3 \cdot 5 \cdot 13
Sign: −0.981+0.193i-0.981 + 0.193i
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), −0.981+0.193i)(2,\ 1560,\ (\ :1/2),\ -0.981 + 0.193i)

Particular Values

L(1)L(1) ≈\approx 0.70384448520.7038444852
L(12)L(\frac12) ≈\approx 0.70384448520.7038444852
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+(−0.432−2.19i)T 1 + (-0.432 - 2.19i)T
13 1−iT 1 - iT
good7 1−7T2 1 - 7T^{2}
11 1+2.86T+11T2 1 + 2.86T + 11T^{2}
17 1−5.52iT−17T2 1 - 5.52iT - 17T^{2}
19 1+3.52T+19T2 1 + 3.52T + 19T^{2}
23 1+7.52iT−23T2 1 + 7.52iT - 23T^{2}
29 1+6.77T+29T2 1 + 6.77T + 29T^{2}
31 1−5.72T+31T2 1 - 5.72T + 31T^{2}
37 1+3.72iT−37T2 1 + 3.72iT - 37T^{2}
41 1+10.1T+41T2 1 + 10.1T + 41T^{2}
43 1−5.52iT−43T2 1 - 5.52iT - 43T^{2}
47 1−8.65iT−47T2 1 - 8.65iT - 47T^{2}
53 1−6.77iT−53T2 1 - 6.77iT - 53T^{2}
59 1+0.593T+59T2 1 + 0.593T + 59T^{2}
61 1+5.25T+61T2 1 + 5.25T + 61T^{2}
67 1+10.5iT−67T2 1 + 10.5iT - 67T^{2}
71 1+2.38T+71T2 1 + 2.38T + 71T^{2}
73 1+5.45iT−73T2 1 + 5.45iT - 73T^{2}
79 1+2.47T+79T2 1 + 2.47T + 79T^{2}
83 1+8.11iT−83T2 1 + 8.11iT - 83T^{2}
89 1−14.1T+89T2 1 - 14.1T + 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

−10.07603361051203060104647849857, −9.105170335525612336289032606322, −8.278135692538432201663442567797, −7.53319639947653276167057760273, −6.41076354955591419663032683115, −5.98888509418124426703172247845, −4.78269923429936598028536343139, −3.94731023187876966275697102425, −2.92970900312235734574531812403, −2.02484172789241170997944818127, 0.25651970391370069379619951743, 1.61383289075840042996961143375, 2.68119735105710948267816707283, 3.91008759763318696290211277627, 5.22313858781190639101729956970, 5.41203491320019935657354033754, 6.67206910552901408641580288668, 7.50358735574620922758862165661, 8.201897937956572162333721948986, 8.923901756996164232770234598811

Graph of the ZZ-function along the critical line