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Properties

Label 2-3549-273.107-c0-0-3
Degree 22
Conductor 35493549
Sign 0.890+0.455i0.890 + 0.455i
Analytic cond. 1.771181.77118
Root an. cond. 1.330851.33085
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  − i·2-s + (0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (0.866 − 0.5i)14-s + (0.499 + 0.866i)15-s − 16-s − i·17-s + (0.866 − 0.499i)18-s + 0.999i·21-s − i·23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  − i·2-s + (0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (0.866 − 0.5i)14-s + (0.499 + 0.866i)15-s − 16-s − i·17-s + (0.866 − 0.499i)18-s + 0.999i·21-s − i·23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

Λ(s)=(3549s/2 ΓC(s) L(s)=((0.890+0.455i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3549s/2 ΓC(s) L(s)=((0.890+0.455i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35493549    =    3⋅7⋅1323 \cdot 7 \cdot 13^{2}
Sign: 0.890+0.455i0.890 + 0.455i
Analytic conductor: 1.771181.77118
Root analytic conductor: 1.330851.33085
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3549(653,⋅)\chi_{3549} (653, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3549, ( :0), 0.890+0.455i)(2,\ 3549,\ (\ :0),\ 0.890 + 0.455i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.3541636262.354163626
L(12)L(\frac12) ≈\approx 2.3541636262.354163626
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
7 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
13 1 1
good2 1+iT−T2 1 + iT - T^{2}
5 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+iT−T2 1 + iT - T^{2}
19 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
23 1+iT−T2 1 + iT - T^{2}
29 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
31 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
37 1+T+T2 1 + T + T^{2}
41 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
43 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
47 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
59 1+iT−T2 1 + iT - T^{2}
61 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
73 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+iT−T2 1 + iT - T^{2}
97 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)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

−8.991467412249484123187020495466, −8.164161465233552998286858413943, −7.25029354575051837693259463267, −6.50272159461540931910327404944, −5.52457752889568522291791341764, −4.72826686179399511543001592120, −3.72600083861810348519052152335, −2.82727628237077085975365676158, −2.34852067084276385749236677723, −1.64386588921025527059791694418, 1.55626075593631418698067638373, 1.96861395291925751262698103846, 3.35928859827508056602516124980, 4.22445639357641663119973548974, 5.38620608056053054488089033824, 5.79685594964412561324847151107, 6.89762976894661481095198282728, 7.21356198534970528034432074353, 7.969545416473219135866105580918, 8.642301791507376077195996862089

Graph of the ZZ-function along the critical line