↑

Properties

Label 2-3960-1320.1019-c0-0-4
Degree 22
Conductor 39603960
Sign 0.994−0.100i0.994 - 0.100i
Analytic cond. 1.976291.97629
Root an. cond. 1.405801.40580
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (1.69 − 0.550i)7-s + (0.309 − 0.951i)8-s − i·10-s + (0.453 − 0.891i)11-s + (−1.16 + 1.59i)13-s + (−1.69 − 0.550i)14-s + (−0.809 + 0.587i)16-s + (1.53 + 0.5i)19-s + (−0.587 + 0.809i)20-s + (−0.891 + 0.453i)22-s + 0.618i·23-s + (−0.309 + 0.951i)25-s + (1.87 − 0.610i)26-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.587 + 0.809i)5-s + (1.69 − 0.550i)7-s + (0.309 − 0.951i)8-s − i·10-s + (0.453 − 0.891i)11-s + (−1.16 + 1.59i)13-s + (−1.69 − 0.550i)14-s + (−0.809 + 0.587i)16-s + (1.53 + 0.5i)19-s + (−0.587 + 0.809i)20-s + (−0.891 + 0.453i)22-s + 0.618i·23-s + (−0.309 + 0.951i)25-s + (1.87 − 0.610i)26-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((0.994−0.100i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s) L(s)=((0.994−0.100i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.994−0.100i0.994 - 0.100i
Analytic conductor: 1.976291.97629
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3960(2339,⋅)\chi_{3960} (2339, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :0), 0.994−0.100i)(2,\ 3960,\ (\ :0),\ 0.994 - 0.100i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2224872131.222487213
L(12)L(\frac12) ≈\approx 1.2224872131.222487213
L(1)L(1) 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+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
3 1 1
5 1+(−0.587−0.809i)T 1 + (-0.587 - 0.809i)T
11 1+(−0.453+0.891i)T 1 + (-0.453 + 0.891i)T
good7 1+(−1.69+0.550i)T+(0.809−0.587i)T2 1 + (-1.69 + 0.550i)T + (0.809 - 0.587i)T^{2}
13 1+(1.16−1.59i)T+(−0.309−0.951i)T2 1 + (1.16 - 1.59i)T + (-0.309 - 0.951i)T^{2}
17 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
19 1+(−1.53−0.5i)T+(0.809+0.587i)T2 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2}
23 1−0.618iT−T2 1 - 0.618iT - T^{2}
29 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
31 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
37 1+(0.0966+0.297i)T+(−0.809+0.587i)T2 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2}
41 1+(−0.280+0.863i)T+(−0.809−0.587i)T2 1 + (-0.280 + 0.863i)T + (-0.809 - 0.587i)T^{2}
43 1+T2 1 + T^{2}
47 1+(1.11+0.363i)T+(0.809+0.587i)T2 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2}
53 1+(1.11−1.53i)T+(−0.309−0.951i)T2 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2}
59 1+(0.297−0.0966i)T+(0.809−0.587i)T2 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2}
61 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
67 1−T2 1 - T^{2}
71 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
73 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
79 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
83 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
89 1+0.907iT−T2 1 + 0.907iT - T^{2}
97 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{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

−8.801543038162235391987793566959, −7.81230172566757876954732463283, −7.39922059789527228806785959730, −6.78298247097957042208455760970, −5.71442821493193925798521400871, −4.75274685756813699226335606337, −3.90688002200632238424936566068, −2.99788156304597302182762857272, −1.91094678613372930944682346103, −1.39260354073603368444943570862, 1.03740206028114100616904381345, 1.87316727869945115540589327442, 2.74505319230986485198041354632, 4.68283745086015150727326450860, 5.03013896388662230735974985692, 5.46240523058045060012687780719, 6.48318354553104386425696098815, 7.49193627508750425941811139937, 7.953666272693591846515477156526, 8.450656966905142275105647565388

Graph of the ZZ-function along the critical line