Properties

Label 2-168e2-1.1-c1-0-115
Degree $2$
Conductor $28224$
Sign $1$
Analytic cond. $225.369$
Root an. cond. $15.0123$
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  + 4·5-s + 6·11-s + 5·13-s + 2·17-s + 19-s + 6·23-s + 11·25-s + 3·31-s − 3·37-s − 6·41-s − 5·43-s − 4·47-s − 6·53-s + 24·55-s + 6·59-s − 2·61-s + 20·65-s − 7·67-s − 16·71-s + 3·73-s + 11·79-s − 12·83-s + 8·85-s + 4·89-s + 4·95-s + 6·97-s + 101-s + ⋯
L(s)  = 1  + 1.78·5-s + 1.80·11-s + 1.38·13-s + 0.485·17-s + 0.229·19-s + 1.25·23-s + 11/5·25-s + 0.538·31-s − 0.493·37-s − 0.937·41-s − 0.762·43-s − 0.583·47-s − 0.824·53-s + 3.23·55-s + 0.781·59-s − 0.256·61-s + 2.48·65-s − 0.855·67-s − 1.89·71-s + 0.351·73-s + 1.23·79-s − 1.31·83-s + 0.867·85-s + 0.423·89-s + 0.410·95-s + 0.609·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(225.369\)
Root analytic conductor: \(15.0123\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{28224} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.229130429\)
\(L(\frac12)\) \(\approx\) \(5.229130429\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.00496190468941, −14.62315016314318, −13.97522224066085, −13.76511757639923, −13.17450848418846, −12.73939775059165, −11.93974280162155, −11.45372470516499, −10.90768004825097, −10.13904689014184, −9.896054634543266, −9.073391318673385, −8.913768060114823, −8.350356623537555, −7.238197371447936, −6.716607432872736, −6.184400716436095, −5.903723124582465, −5.077486102344199, −4.522713415387706, −3.451638653801799, −3.193576913589489, −2.042342432244802, −1.407790056497197, −1.043406576644783, 1.043406576644783, 1.407790056497197, 2.042342432244802, 3.193576913589489, 3.451638653801799, 4.522713415387706, 5.077486102344199, 5.903723124582465, 6.184400716436095, 6.716607432872736, 7.238197371447936, 8.350356623537555, 8.913768060114823, 9.073391318673385, 9.896054634543266, 10.13904689014184, 10.90768004825097, 11.45372470516499, 11.93974280162155, 12.73939775059165, 13.17450848418846, 13.76511757639923, 13.97522224066085, 14.62315016314318, 15.00496190468941

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