↑

Properties

Label 2-3840-40.29-c1-0-42
Degree 22
Conductor 38403840
Sign 0.948+0.316i0.948 + 0.316i
Analytic cond. 30.662530.6625
Root an. cond. 5.537375.53737
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  − 3-s + (1 + 2i)5-s − 2i·7-s + 9-s + 2i·11-s − 6·13-s + (−1 − 2i)15-s − 2i·17-s + 2i·21-s − 4i·23-s + (−3 + 4i)25-s − 27-s + 8·31-s − 2i·33-s + (4 − 2i)35-s + ⋯
L(s)  = 1  − 0.577·3-s + (0.447 + 0.894i)5-s − 0.755i·7-s + 0.333·9-s + 0.603i·11-s − 1.66·13-s + (−0.258 − 0.516i)15-s − 0.485i·17-s + 0.436i·21-s − 0.834i·23-s + (−0.600 + 0.800i)25-s − 0.192·27-s + 1.43·31-s − 0.348i·33-s + (0.676 − 0.338i)35-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=((0.948+0.316i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s+1/2) L(s)=((0.948+0.316i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: 0.948+0.316i0.948 + 0.316i
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3840(2689,⋅)\chi_{3840} (2689, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3840, ( :1/2), 0.948+0.316i)(2,\ 3840,\ (\ :1/2),\ 0.948 + 0.316i)

Particular Values

L(1)L(1) ≈\approx 1.3269444361.326944436
L(12)L(\frac12) ≈\approx 1.3269444361.326944436
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+T 1 + T
5 1+(−1−2i)T 1 + (-1 - 2i)T
good7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1−2iT−11T2 1 - 2iT - 11T^{2}
13 1+6T+13T2 1 + 6T + 13T^{2}
17 1+2iT−17T2 1 + 2iT - 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+8iT−47T2 1 + 8iT - 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+10iT−59T2 1 + 10iT - 59T^{2}
61 1−2iT−61T2 1 - 2iT - 61T^{2}
67 1−8T+67T2 1 - 8T + 67T^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1−4iT−73T2 1 - 4iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1−8iT−97T2 1 - 8iT - 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

−8.320474709041013846904569692056, −7.37435437745216037539003831738, −7.00391833884896956190005850099, −6.44437738256167999445675951038, −5.37891102645110930637204927089, −4.76392071969394491282135993248, −3.93149578012733630907591592964, −2.75999272539017993770504633101, −2.07046516807725488064017094206, −0.56198214387489170079528983524, 0.809691506752474739841906517454, 1.99012172003255131090625225102, 2.85210407774094641046806171620, 4.16665499899804697515247924569, 4.92183341417785139541281227482, 5.54262494717769131249850345506, 6.06467204441022122435988759374, 7.00074252106946386729288262558, 7.87976864954381779992624312889, 8.553581912343736601123640740697

Graph of the ZZ-function along the critical line