↑

Properties

Label 2-3120-5.4-c1-0-16
Degree 22
Conductor 31203120
Sign −0.193−0.981i-0.193 - 0.981i
Analytic cond. 24.913324.9133
Root an. cond. 4.991324.99132
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)=(3120s/2 ΓC(s) L(s)=((−0.193−0.981i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3120s/2 ΓC(s+1/2) L(s)=((−0.193−0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31203120    =    24⋅3⋅5⋅132^{4} \cdot 3 \cdot 5 \cdot 13
Sign: −0.193−0.981i-0.193 - 0.981i
Analytic conductor: 24.913324.9133
Root analytic conductor: 4.991324.99132
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3120(1249,⋅)\chi_{3120} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3120, ( :1/2), −0.193−0.981i)(2,\ 3120,\ (\ :1/2),\ -0.193 - 0.981i)

Particular Values

L(1)L(1) ≈\approx 1.4219255461.421925546
L(12)L(\frac12) ≈\approx 1.4219255461.421925546
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

−9.002442362749210980285217041599, −7.974736553810754390217886047870, −7.24682617983848337553944760922, −6.82781925388679212569334095157, −5.87283293147336683891715076947, −5.42747770206425852654020393481, −3.77085828579888510112810546576, −3.55415593881964060690529633781, −2.14746807883972408889166258360, −1.47437469455835327633122598897, 0.43442245076966229069987308226, 1.64811885978877486401605361282, 2.90570967842718281498014123252, 3.84284258198109916668165331117, 4.71502003912366397323264245244, 5.22611423518205722776456488711, 6.08326113847345923609149590202, 6.98023794501156461579400304303, 7.86385346803128622375795995910, 8.653960438816911146143198554705

Graph of the ZZ-function along the critical line