Properties

Label 2-168e2-1.1-c1-0-123
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 5·11-s + 2·13-s − 6·17-s − 2·19-s + 6·23-s − 4·25-s + 3·29-s + 5·31-s + 2·37-s − 8·41-s − 4·43-s + 4·47-s − 9·53-s − 5·55-s + 3·59-s − 12·61-s − 2·65-s + 2·67-s + 8·71-s + 14·73-s − 79-s − 17·83-s + 6·85-s − 18·89-s + 2·95-s − 3·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.50·11-s + 0.554·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s − 4/5·25-s + 0.557·29-s + 0.898·31-s + 0.328·37-s − 1.24·41-s − 0.609·43-s + 0.583·47-s − 1.23·53-s − 0.674·55-s + 0.390·59-s − 1.53·61-s − 0.248·65-s + 0.244·67-s + 0.949·71-s + 1.63·73-s − 0.112·79-s − 1.86·83-s + 0.650·85-s − 1.90·89-s + 0.205·95-s − 0.304·97-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: \(1\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−15.34836451182658, −15.23169866385875, −14.34253322217217, −13.88814726596658, −13.44140060476921, −12.78542461467251, −12.25246496474984, −11.61734167808948, −11.26260544238567, −10.82795591326938, −10.03119220788415, −9.411097095951107, −8.905385891570894, −8.437963894203811, −7.908288433889937, −6.917133351480515, −6.667285081294579, −6.204526697226877, −5.289782764305487, −4.493298045578080, −4.153663963065205, −3.450138091249182, −2.712255674380128, −1.769491303780047, −1.093419670973865, 0, 1.093419670973865, 1.769491303780047, 2.712255674380128, 3.450138091249182, 4.153663963065205, 4.493298045578080, 5.289782764305487, 6.204526697226877, 6.667285081294579, 6.917133351480515, 7.908288433889937, 8.437963894203811, 8.905385891570894, 9.411097095951107, 10.03119220788415, 10.82795591326938, 11.26260544238567, 11.61734167808948, 12.25246496474984, 12.78542461467251, 13.44140060476921, 13.88814726596658, 14.34253322217217, 15.23169866385875, 15.34836451182658

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