Properties

Label 2-168e2-1.1-c1-0-116
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 − 3·11-s + 6·13-s + 5·17-s + 19-s − 7·23-s − 4·25-s + 2·29-s + 5·31-s − 3·37-s + 2·41-s − 4·43-s + 5·47-s − 53-s + 3·55-s − 15·59-s + 5·61-s − 6·65-s − 9·67-s + 7·73-s − 79-s − 12·83-s − 5·85-s − 7·89-s − 95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.904·11-s + 1.66·13-s + 1.21·17-s + 0.229·19-s − 1.45·23-s − 4/5·25-s + 0.371·29-s + 0.898·31-s − 0.493·37-s + 0.312·41-s − 0.609·43-s + 0.729·47-s − 0.137·53-s + 0.404·55-s − 1.95·59-s + 0.640·61-s − 0.744·65-s − 1.09·67-s + 0.819·73-s − 0.112·79-s − 1.31·83-s − 0.542·85-s − 0.741·89-s − 0.102·95-s − 0.203·97-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: \(1\)
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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
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

−15.62674326691225, −15.17582546011505, −14.16682806987910, −13.92992106035140, −13.50010829696929, −12.73537853281610, −12.29855323126374, −11.67071837612174, −11.31809004282823, −10.47086140713671, −10.24239789598033, −9.608480520751862, −8.787340756131875, −8.267546828437400, −7.840652720746090, −7.406257518249888, −6.424004373672975, −5.959131708900274, −5.487417130685759, −4.644328764599759, −3.957128086376755, −3.430817034620115, −2.778750426799052, −1.794353409647582, −1.043725912036251, 0, 1.043725912036251, 1.794353409647582, 2.778750426799052, 3.430817034620115, 3.957128086376755, 4.644328764599759, 5.487417130685759, 5.959131708900274, 6.424004373672975, 7.406257518249888, 7.840652720746090, 8.267546828437400, 8.787340756131875, 9.608480520751862, 10.24239789598033, 10.47086140713671, 11.31809004282823, 11.67071837612174, 12.29855323126374, 12.73537853281610, 13.50010829696929, 13.92992106035140, 14.16682806987910, 15.17582546011505, 15.62674326691225

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