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Properties

Label 2-448-1.1-c5-0-28
Degree 22
Conductor 448448
Sign 11
Analytic cond. 71.851971.8519
Root an. cond. 8.476558.47655
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 30·3-s − 32·5-s − 49·7-s + 657·9-s − 624·11-s + 708·13-s − 960·15-s + 934·17-s + 1.85e3·19-s − 1.47e3·21-s + 1.12e3·23-s − 2.10e3·25-s + 1.24e4·27-s + 1.17e3·29-s − 2.90e3·31-s − 1.87e4·33-s + 1.56e3·35-s + 1.24e4·37-s + 2.12e4·39-s + 2.66e3·41-s − 7.14e3·43-s − 2.10e4·45-s + 7.46e3·47-s + 2.40e3·49-s + 2.80e4·51-s + 2.72e4·53-s + 1.99e4·55-s + ⋯
L(s)  = 1  + 1.92·3-s − 0.572·5-s − 0.377·7-s + 2.70·9-s − 1.55·11-s + 1.16·13-s − 1.10·15-s + 0.783·17-s + 1.18·19-s − 0.727·21-s + 0.441·23-s − 0.672·25-s + 3.27·27-s + 0.259·29-s − 0.543·31-s − 2.99·33-s + 0.216·35-s + 1.49·37-s + 2.23·39-s + 0.247·41-s − 0.589·43-s − 1.54·45-s + 0.493·47-s + 1/7·49-s + 1.50·51-s + 1.33·53-s + 0.890·55-s + ⋯

Functional equation

Λ(s)=(448s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(448s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 448448    =    26⋅72^{6} \cdot 7
Sign: 11
Analytic conductor: 71.851971.8519
Root analytic conductor: 8.476558.47655
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 448, ( :5/2), 1)(2,\ 448,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 4.1875901504.187590150
L(12)L(\frac12) ≈\approx 4.1875901504.187590150
L(72)L(\frac{7}{2}) 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
7 1+p2T 1 + p^{2} T
good3 1−10pT+p5T2 1 - 10 p T + p^{5} T^{2}
5 1+32T+p5T2 1 + 32 T + p^{5} T^{2}
11 1+624T+p5T2 1 + 624 T + p^{5} T^{2}
13 1−708T+p5T2 1 - 708 T + p^{5} T^{2}
17 1−934T+p5T2 1 - 934 T + p^{5} T^{2}
19 1−1858T+p5T2 1 - 1858 T + p^{5} T^{2}
23 1−1120T+p5T2 1 - 1120 T + p^{5} T^{2}
29 1−1174T+p5T2 1 - 1174 T + p^{5} T^{2}
31 1+2908T+p5T2 1 + 2908 T + p^{5} T^{2}
37 1−12462T+p5T2 1 - 12462 T + p^{5} T^{2}
41 1−2662T+p5T2 1 - 2662 T + p^{5} T^{2}
43 1+7144T+p5T2 1 + 7144 T + p^{5} T^{2}
47 1−7468T+p5T2 1 - 7468 T + p^{5} T^{2}
53 1−27274T+p5T2 1 - 27274 T + p^{5} T^{2}
59 1−2490T+p5T2 1 - 2490 T + p^{5} T^{2}
61 1−11096T+p5T2 1 - 11096 T + p^{5} T^{2}
67 1−39756T+p5T2 1 - 39756 T + p^{5} T^{2}
71 1−69888T+p5T2 1 - 69888 T + p^{5} T^{2}
73 1−16450T+p5T2 1 - 16450 T + p^{5} T^{2}
79 1+78376T+p5T2 1 + 78376 T + p^{5} T^{2}
83 1−109818T+p5T2 1 - 109818 T + p^{5} T^{2}
89 1+56966T+p5T2 1 + 56966 T + p^{5} T^{2}
97 1+115946T+p5T2 1 + 115946 T + p^{5} 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

−9.996118724105065156668466334681, −9.345043319960076057262833342846, −8.230570626432969136775843943954, −7.900777959191941085034757036604, −7.03178155852422763538641808158, −5.43543856737362525337529338711, −4.00768697279557450238735317728, −3.27488949204659806931351649637, −2.45631589271709401571224111849, −1.00755590478842028016131675157, 1.00755590478842028016131675157, 2.45631589271709401571224111849, 3.27488949204659806931351649637, 4.00768697279557450238735317728, 5.43543856737362525337529338711, 7.03178155852422763538641808158, 7.900777959191941085034757036604, 8.230570626432969136775843943954, 9.345043319960076057262833342846, 9.996118724105065156668466334681

Graph of the ZZ-function along the critical line