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Properties

Label 2-3072-48.29-c0-0-8
Degree 22
Conductor 30723072
Sign −0.382+0.923i-0.382 + 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+0.923i-0.382 + 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 0.73044162200.7304416220
L(12)L(\frac12) ≈\approx 0.73044162200.7304416220
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

−8.624354187827936717032718184389, −7.79040635287870126992935266952, −7.05304989496546062757050854351, −6.42785414255528850372055491136, −5.75452046609404865511276035549, −4.81478377608992146015288375312, −4.16230907033899008918432432400, −2.72425410088202730598982268643, −1.95712338329204378325968463816, −0.51063105119599922290331032342, 1.36725538656361626900124952109, 2.74198755636911149032372621669, 3.89702041163973258740634789180, 4.32620683929509160746485781991, 5.43063973870499996027286541794, 5.94094145167146399279470691750, 6.71287564274634971063025448154, 7.61252048033858873517136605199, 8.501288829706252621221386772923, 9.188777729223045129511174371523

Graph of the ZZ-function along the critical line