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Properties

Label 2-57e2-19.18-c0-0-1
Degree 22
Conductor 32493249
Sign 0.917+0.397i0.917 + 0.397i
Analytic cond. 1.621461.62146
Root an. cond. 1.273361.27336
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  + 4-s + 7-s − 1.73i·13-s + 16-s − 25-s + 28-s − 1.73i·31-s + 1.73i·37-s − 43-s − 1.73i·52-s + 61-s + 64-s + 1.73i·67-s − 73-s + 1.73i·79-s + ⋯
L(s)  = 1  + 4-s + 7-s − 1.73i·13-s + 16-s − 25-s + 28-s − 1.73i·31-s + 1.73i·37-s − 43-s − 1.73i·52-s + 61-s + 64-s + 1.73i·67-s − 73-s + 1.73i·79-s + ⋯

Functional equation

Λ(s)=(3249s/2 ΓC(s) L(s)=((0.917+0.397i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3249s/2 ΓC(s) L(s)=((0.917+0.397i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32493249    =    32⋅1923^{2} \cdot 19^{2}
Sign: 0.917+0.397i0.917 + 0.397i
Analytic conductor: 1.621461.62146
Root analytic conductor: 1.273361.27336
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3249(721,⋅)\chi_{3249} (721, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3249, ( :0), 0.917+0.397i)(2,\ 3249,\ (\ :0),\ 0.917 + 0.397i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7945230581.794523058
L(12)L(\frac12) ≈\approx 1.7945230581.794523058
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)
bad3 1 1
19 1 1
good2 1−T2 1 - T^{2}
5 1+T2 1 + T^{2}
7 1−T+T2 1 - T + T^{2}
11 1+T2 1 + T^{2}
13 1+1.73iT−T2 1 + 1.73iT - T^{2}
17 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
29 1−T2 1 - T^{2}
31 1+1.73iT−T2 1 + 1.73iT - T^{2}
37 1−1.73iT−T2 1 - 1.73iT - T^{2}
41 1−T2 1 - T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+T2 1 + T^{2}
53 1−T2 1 - T^{2}
59 1−T2 1 - T^{2}
61 1−T+T2 1 - T + T^{2}
67 1−1.73iT−T2 1 - 1.73iT - T^{2}
71 1−T2 1 - T^{2}
73 1+T+T2 1 + T + T^{2}
79 1−1.73iT−T2 1 - 1.73iT - T^{2}
83 1+T2 1 + 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

−8.393072788487918333246616729286, −8.030632088042942430648472805571, −7.46052676690528089544895530621, −6.52310175908521593121712440784, −5.71195110835093055165030796696, −5.17487453540568805437357798884, −4.04602307569982952741835905998, −3.05578334199597412486305655856, −2.24075731069851620029547630904, −1.18373433663797589396647250814, 1.63304827113624410040614517528, 2.03366202953463195198326914369, 3.30135493110515896945165933230, 4.24226939882013055789386143829, 5.06936897426915465335118413157, 5.95235361466803400733115531047, 6.76285230170488720401171019306, 7.29228633293852951038404810286, 8.066762217316811310599025086328, 8.812151473651458743572696761973

Graph of the ZZ-function along the critical line