Properties

Label 2-72128-1.1-c1-0-0
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

\(L(1)\) \(\approx\) \(0.5167571355\)
\(L(\frac12)\) \(\approx\) \(0.5167571355\)
\(L(\frac{3}{2})\) 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

−14.19565197641503, −13.41317502043466, −13.12995022321050, −13.00959474711753, −12.26371576024906, −11.41289132186810, −11.03749270630361, −10.70853587269509, −9.899546089221611, −9.464208446888673, −8.999386743250727, −8.421123261890993, −8.128844920624402, −7.385255611022206, −7.233175844068961, −6.142447329274343, −5.907570275314556, −5.137176377011057, −4.359325820386234, −3.991166729009377, −3.332885687891816, −2.516550081091296, −2.228844167110805, −1.697819305558377, −0.1965956170535469, 0.1965956170535469, 1.697819305558377, 2.228844167110805, 2.516550081091296, 3.332885687891816, 3.991166729009377, 4.359325820386234, 5.137176377011057, 5.907570275314556, 6.142447329274343, 7.233175844068961, 7.385255611022206, 8.128844920624402, 8.421123261890993, 8.999386743250727, 9.464208446888673, 9.899546089221611, 10.70853587269509, 11.03749270630361, 11.41289132186810, 12.26371576024906, 13.00959474711753, 13.12995022321050, 13.41317502043466, 14.19565197641503

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