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Properties

Label 2-3528-168.59-c0-0-0
Degree 22
Conductor 35283528
Sign 0.726+0.686i0.726 + 0.686i
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 + 0.707i)8-s + (1.22 + 0.707i)11-s + (0.500 + 0.866i)16-s + (1 − 0.999i)22-s + (0.707 + 1.22i)23-s + (−0.5 + 0.866i)25-s + 1.41·29-s + (0.965 − 0.258i)32-s + (−1.73 + i)37-s + (−0.707 − 1.22i)44-s + (1.36 − 0.366i)46-s + (0.707 + 0.707i)50-s + (0.707 − 1.22i)53-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 + 0.707i)8-s + (1.22 + 0.707i)11-s + (0.500 + 0.866i)16-s + (1 − 0.999i)22-s + (0.707 + 1.22i)23-s + (−0.5 + 0.866i)25-s + 1.41·29-s + (0.965 − 0.258i)32-s + (−1.73 + i)37-s + (−0.707 − 1.22i)44-s + (1.36 − 0.366i)46-s + (0.707 + 0.707i)50-s + (0.707 − 1.22i)53-s + ⋯

Functional equation

Λ(s)=(3528s/2 ΓC(s) L(s)=((0.726+0.686i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3528s/2 ΓC(s) L(s)=((0.726+0.686i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35283528    =    23⋅32⋅722^{3} \cdot 3^{2} \cdot 7^{2}
Sign: 0.726+0.686i0.726 + 0.686i
Analytic conductor: 1.760701.76070
Root analytic conductor: 1.326911.32691
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3528(1403,⋅)\chi_{3528} (1403, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3528, ( :0), 0.726+0.686i)(2,\ 3528,\ (\ :0),\ 0.726 + 0.686i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3802225211.380222521
L(12)L(\frac12) ≈\approx 1.3802225211.380222521
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+(−0.258+0.965i)T 1 + (-0.258 + 0.965i)T
3 1 1
7 1 1
good5 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
13 1+T2 1 + T^{2}
17 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(−0.707−1.22i)T+(−0.5+0.866i)T2 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2}
29 1−1.41T+T2 1 - 1.41T + T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1+(1.73−i)T+(0.5−0.866i)T2 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1+T2 1 + T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(−0.707+1.22i)T+(−0.5−0.866i)T2 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}
59 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
61 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
67 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
71 1−1.41T+T2 1 - 1.41T + T^{2}
73 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)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

−8.884555574498939132497389245930, −8.136196113736627041831518184045, −7.05569232645121408037593434522, −6.43562764158603928984961105537, −5.34700900326431833109729184901, −4.79673297294158031673034203932, −3.79875582812275682665550535434, −3.26899926929455470983045365633, −2.01885182563312885816428248709, −1.23844819419050847025256862902, 0.904979083194529587460774075689, 2.58966208542478317743268166827, 3.64685249252020382249505325622, 4.28117733334761202681365142535, 5.14116651855249354344358343557, 5.99713625458241546657323341157, 6.62475416398687705227348808946, 7.12434524397168184543890269904, 8.225596442969306602008673647371, 8.650139706603085326025513958949

Graph of the ZZ-function along the critical line