Properties

Label 2-59248-1.1-c1-0-0
Degree $2$
Conductor $59248$
Sign $1$
Analytic cond. $473.097$
Root an. cond. $21.7508$
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 + 7-s + 9-s + 4·11-s − 6·17-s − 6·19-s − 2·21-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 + 49-s + 12·51-s + 6·53-s + 12·57-s + 2·59-s + 63-s + 8·71-s − 6·73-s + 10·75-s + 4·77-s − 8·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.45·17-s − 1.37·19-s − 0.436·21-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 + 1/7·49-s + 1.68·51-s + 0.824·53-s + 1.58·57-s + 0.260·59-s + 0.125·63-s + 0.949·71-s − 0.702·73-s + 1.15·75-s + 0.455·77-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(59248\)    =    \(2^{4} \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(473.097\)
Root analytic conductor: \(21.7508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{59248} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 59248,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6089543925\)
\(L(\frac12)\) \(\approx\) \(0.6089543925\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 \)
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

−14.29077796608307, −13.88458959804547, −13.16596423350261, −12.81518591958445, −12.08282921262579, −11.71092900783473, −11.36054810919544, −10.96177931450693, −10.18464787987826, −10.02430724847635, −9.058218263641016, −8.549215904275185, −8.372503967396231, −7.351320503608160, −6.719252564654405, −6.365936773755588, −6.109363698113635, −5.133243556013725, −4.791959286252853, −4.194325371371835, −3.660371763974795, −2.668316613812702, −1.918131678642109, −1.314733814139281, −0.2944196793963340, 0.2944196793963340, 1.314733814139281, 1.918131678642109, 2.668316613812702, 3.660371763974795, 4.194325371371835, 4.791959286252853, 5.133243556013725, 6.109363698113635, 6.365936773755588, 6.719252564654405, 7.351320503608160, 8.372503967396231, 8.549215904275185, 9.058218263641016, 10.02430724847635, 10.18464787987826, 10.96177931450693, 11.36054810919544, 11.71092900783473, 12.08282921262579, 12.81518591958445, 13.16596423350261, 13.88458959804547, 14.29077796608307

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