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Properties

Label 2-3528-3.2-c0-0-0
Degree 22
Conductor 35283528
Sign −0.816−0.577i-0.816 - 0.577i
Analytic cond. 1.760701.76070
Root an. cond. 1.326911.32691
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·5-s + 1.41i·11-s − 13-s − 19-s − 1.00·25-s − 31-s + 37-s + 1.41i·41-s − 43-s − 1.41i·47-s − 2.00·55-s − 1.41i·65-s − 67-s − 1.41i·71-s + 73-s + ⋯
L(s)  = 1  + 1.41i·5-s + 1.41i·11-s − 13-s − 19-s − 1.00·25-s − 31-s + 37-s + 1.41i·41-s − 43-s − 1.41i·47-s − 2.00·55-s − 1.41i·65-s − 67-s − 1.41i·71-s + 73-s + ⋯

Functional equation

Λ(s)=(3528s/2 ΓC(s) L(s)=((−0.816−0.577i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3528s/2 ΓC(s) L(s)=((−0.816−0.577i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35283528    =    23⋅32⋅722^{3} \cdot 3^{2} \cdot 7^{2}
Sign: −0.816−0.577i-0.816 - 0.577i
Analytic conductor: 1.760701.76070
Root analytic conductor: 1.326911.32691
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3528(1961,⋅)\chi_{3528} (1961, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3528, ( :0), −0.816−0.577i)(2,\ 3528,\ (\ :0),\ -0.816 - 0.577i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.83498957060.8349895706
L(12)L(\frac12) ≈\approx 0.83498957060.8349895706
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 1
3 1 1
7 1 1
good5 1−1.41iT−T2 1 - 1.41iT - T^{2}
11 1−1.41iT−T2 1 - 1.41iT - T^{2}
13 1+T+T2 1 + T + T^{2}
17 1−T2 1 - T^{2}
19 1+T+T2 1 + T + T^{2}
23 1−T2 1 - T^{2}
29 1−T2 1 - T^{2}
31 1+T+T2 1 + T + T^{2}
37 1−T+T2 1 - T + T^{2}
41 1−1.41iT−T2 1 - 1.41iT - T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+1.41iT−T2 1 + 1.41iT - T^{2}
53 1−T2 1 - T^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1+T+T2 1 + T + T^{2}
71 1+1.41iT−T2 1 + 1.41iT - T^{2}
73 1−T+T2 1 - T + T^{2}
79 1−T+T2 1 - T + T^{2}
83 1−1.41iT−T2 1 - 1.41iT - T^{2}
89 1−T2 1 - T^{2}
97 1+T2 1 + T^{2}
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   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

−9.249313022419241038025943340773, −8.071865626468637297677640735748, −7.47276049556730283720575287328, −6.80349605717747208566743382481, −6.37456668973445692327809245241, −5.20220034525360588401754815157, −4.45604792239740404381996476053, −3.52989060463489979234613393059, −2.54109836783216901845155411096, −1.95146520649301826951840366306, 0.45868234328069865893421138394, 1.69635927657562640312497724338, 2.83693049799606661499954259705, 3.93163565521617200024834074640, 4.65674984675802056763699125099, 5.43540895051689889280931443875, 6.00731598375258856756709413835, 7.01577948241989213884263781450, 7.946776818528765003728186015330, 8.484203178123183192353498170526

Graph of the ZZ-function along the critical line