Properties

Label 2-168e2-1.1-c1-0-107
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  − 2·5-s + 2·11-s − 4·13-s + 6·17-s + 8·19-s + 6·23-s − 25-s − 10·29-s − 4·31-s − 6·37-s − 6·41-s − 4·43-s + 8·47-s + 2·53-s − 4·55-s + 4·59-s − 8·61-s + 8·65-s + 8·67-s + 10·71-s − 4·73-s + 4·79-s − 12·83-s − 12·85-s − 14·89-s − 16·95-s − 4·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.603·11-s − 1.10·13-s + 1.45·17-s + 1.83·19-s + 1.25·23-s − 1/5·25-s − 1.85·29-s − 0.718·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 0.274·53-s − 0.539·55-s + 0.520·59-s − 1.02·61-s + 0.992·65-s + 0.977·67-s + 1.18·71-s − 0.468·73-s + 0.450·79-s − 1.31·83-s − 1.30·85-s − 1.48·89-s − 1.64·95-s − 0.406·97-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−15.42541313541755, −14.92920403772995, −14.53391897136016, −13.94330569673436, −13.43541469568813, −12.57351962495318, −12.27185614619171, −11.73157755532264, −11.36145004874708, −10.73493535125598, −9.850829968884721, −9.629793464518068, −9.013106755040621, −8.279780101342614, −7.603442577607360, −7.240736921144692, −6.949661959293170, −5.706190818477522, −5.379010060173028, −4.817305230306494, −3.785567101623979, −3.523663733960549, −2.845643564642319, −1.766097056566737, −1.003379493031104, 0, 1.003379493031104, 1.766097056566737, 2.845643564642319, 3.523663733960549, 3.785567101623979, 4.817305230306494, 5.379010060173028, 5.706190818477522, 6.949661959293170, 7.240736921144692, 7.603442577607360, 8.279780101342614, 9.013106755040621, 9.629793464518068, 9.850829968884721, 10.73493535125598, 11.36145004874708, 11.73157755532264, 12.27185614619171, 12.57351962495318, 13.43541469568813, 13.94330569673436, 14.53391897136016, 14.92920403772995, 15.42541313541755

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