Properties

Label 2-168e2-1.1-c1-0-11
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  − 5-s + 3·11-s − 4·13-s + 4·19-s − 8·23-s − 4·25-s − 3·29-s − 5·31-s − 8·37-s − 8·41-s − 6·43-s − 10·47-s + 9·53-s − 3·55-s − 5·59-s + 10·61-s + 4·65-s − 6·67-s − 10·71-s + 2·73-s + 11·79-s + 7·83-s + 18·89-s − 4·95-s − 17·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.904·11-s − 1.10·13-s + 0.917·19-s − 1.66·23-s − 4/5·25-s − 0.557·29-s − 0.898·31-s − 1.31·37-s − 1.24·41-s − 0.914·43-s − 1.45·47-s + 1.23·53-s − 0.404·55-s − 0.650·59-s + 1.28·61-s + 0.496·65-s − 0.733·67-s − 1.18·71-s + 0.234·73-s + 1.23·79-s + 0.768·83-s + 1.90·89-s − 0.410·95-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(0.9320264625\)
\(L(\frac12)\) \(\approx\) \(0.9320264625\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−15.09323664136057, −14.72759170992098, −14.24199540845170, −13.57519202147676, −13.25417191333929, −12.20978440115131, −11.97278620275651, −11.76544310977631, −10.96183480609535, −10.24250464447066, −9.748037608224705, −9.383238563897809, −8.573131558733726, −8.068626473182325, −7.436658650502471, −6.990589396694599, −6.349555958558227, −5.564685999876322, −5.088947268847737, −4.307476088599606, −3.648787021678136, −3.250558260889036, −2.053968917189272, −1.674997216649155, −0.3602126966609361, 0.3602126966609361, 1.674997216649155, 2.053968917189272, 3.250558260889036, 3.648787021678136, 4.307476088599606, 5.088947268847737, 5.564685999876322, 6.349555958558227, 6.990589396694599, 7.436658650502471, 8.068626473182325, 8.573131558733726, 9.383238563897809, 9.748037608224705, 10.24250464447066, 10.96183480609535, 11.76544310977631, 11.97278620275651, 12.20978440115131, 13.25417191333929, 13.57519202147676, 14.24199540845170, 14.72759170992098, 15.09323664136057

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