Properties

Label 2-72128-1.1-c1-0-13
Degree $2$
Conductor $72128$
Sign $1$
Analytic cond. $575.944$
Root an. cond. $23.9988$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s + 4·11-s − 6·17-s + 6·19-s + 23-s − 5·25-s + 4·27-s − 10·29-s + 4·31-s − 8·33-s + 2·37-s + 10·41-s − 4·43-s + 12·47-s + 12·51-s + 6·53-s − 12·57-s + 2·59-s − 2·69-s + 8·71-s + 6·73-s + 10·75-s + 8·79-s − 11·81-s + 14·83-s + 20·87-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 1.20·11-s − 1.45·17-s + 1.37·19-s + 0.208·23-s − 25-s + 0.769·27-s − 1.85·29-s + 0.718·31-s − 1.39·33-s + 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.75·47-s + 1.68·51-s + 0.824·53-s − 1.58·57-s + 0.260·59-s − 0.240·69-s + 0.949·71-s + 0.702·73-s + 1.15·75-s + 0.900·79-s − 1.22·81-s + 1.53·83-s + 2.14·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72128\)    =    \(2^{6} \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(575.944\)
Root analytic conductor: \(23.9988\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{72128} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72128,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.561042270\)
\(L(\frac12)\) \(\approx\) \(1.561042270\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
23 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.04137982272777, −13.56359359254162, −13.15652726556076, −12.36282274438664, −12.07581574908783, −11.43850630937217, −11.29566013203328, −10.81863826193371, −10.10856941633669, −9.512203180188110, −9.097491062788087, −8.709328185508334, −7.670494135374324, −7.465819245773855, −6.647978337172334, −6.316660572051982, −5.792430332373190, −5.245130146376600, −4.709032674942474, −3.921673720157691, −3.685352975590370, −2.580537603861506, −2.000814222176713, −1.057211563367311, −0.5276932042206680, 0.5276932042206680, 1.057211563367311, 2.000814222176713, 2.580537603861506, 3.685352975590370, 3.921673720157691, 4.709032674942474, 5.245130146376600, 5.792430332373190, 6.316660572051982, 6.647978337172334, 7.465819245773855, 7.670494135374324, 8.709328185508334, 9.097491062788087, 9.512203180188110, 10.10856941633669, 10.81863826193371, 11.29566013203328, 11.43850630937217, 12.07581574908783, 12.36282274438664, 13.15652726556076, 13.56359359254162, 14.04137982272777

Graph of the $Z$-function along the critical line