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Properties

Label 2-3072-48.29-c0-0-3
Degree 22
Conductor 30723072
Sign 0.382−0.923i0.382 - 0.923i
Analytic cond. 1.533121.53312
Root an. cond. 1.238191.23819
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  + (0.707 + 0.707i)3-s + 1.00i·9-s + (1.41 + 1.41i)19-s − i·25-s + (−0.707 + 0.707i)27-s + (−1.41 + 1.41i)43-s + 49-s + 2.00i·57-s + (1.41 + 1.41i)67-s − 2i·73-s + (0.707 − 0.707i)75-s − 1.00·81-s − 2·97-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + 1.00i·9-s + (1.41 + 1.41i)19-s − i·25-s + (−0.707 + 0.707i)27-s + (−1.41 + 1.41i)43-s + 49-s + 2.00i·57-s + (1.41 + 1.41i)67-s − 2i·73-s + (0.707 − 0.707i)75-s − 1.00·81-s − 2·97-s + ⋯

Functional equation

Λ(s)=(3072s/2 ΓC(s) L(s)=((0.382−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3072s/2 ΓC(s) L(s)=((0.382−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30723072    =    210⋅32^{10} \cdot 3
Sign: 0.382−0.923i0.382 - 0.923i
Analytic conductor: 1.533121.53312
Root analytic conductor: 1.238191.23819
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3072(257,⋅)\chi_{3072} (257, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3072, ( :0), 0.382−0.923i)(2,\ 3072,\ (\ :0),\ 0.382 - 0.923i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6041839661.604183966
L(12)L(\frac12) ≈\approx 1.6041839661.604183966
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good5 1+iT2 1 + iT^{2}
7 1−T2 1 - T^{2}
11 1+iT2 1 + iT^{2}
13 1+iT2 1 + iT^{2}
17 1−T2 1 - T^{2}
19 1+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)T + iT^{2}
23 1+T2 1 + T^{2}
29 1−iT2 1 - iT^{2}
31 1+T2 1 + T^{2}
37 1−iT2 1 - iT^{2}
41 1+T2 1 + T^{2}
43 1+(1.41−1.41i)T−iT2 1 + (1.41 - 1.41i)T - iT^{2}
47 1−T2 1 - T^{2}
53 1+iT2 1 + iT^{2}
59 1+iT2 1 + iT^{2}
61 1+iT2 1 + iT^{2}
67 1+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)T + iT^{2}
71 1+T2 1 + T^{2}
73 1+2iT−T2 1 + 2iT - T^{2}
79 1+T2 1 + T^{2}
83 1−iT2 1 - iT^{2}
89 1+T2 1 + T^{2}
97 1+2T+T2 1 + 2T + 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

−9.073221786252299923806390063414, −8.200111280201559902131618305349, −7.83501986510574242241607440647, −6.87203775467530392545073529481, −5.84708554138472942478538196571, −5.12709580101326776447169180230, −4.24377471230616338001811570280, −3.48088156479517302531132840367, −2.68939251459845200889645947595, −1.53350743937574894770540309698, 0.994218263864976680754827617339, 2.15055032143309626007799534134, 3.06496364151215791319726696955, 3.77662218771910359513202118686, 4.98295731119253467662722568788, 5.70008998277433213533701522031, 6.89726913440852268164912887252, 7.10191604597371451993263308553, 8.001015597769676503500658238449, 8.723918046626110272180542179297

Graph of the ZZ-function along the critical line