Properties

Label 2-168e2-1.1-c1-0-134
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 − 4·11-s + 6·13-s + 2·17-s − 4·19-s − 8·23-s − 25-s − 2·29-s + 10·37-s − 6·41-s + 4·43-s + 6·53-s − 8·55-s − 4·59-s + 6·61-s + 12·65-s − 4·67-s − 8·71-s − 10·73-s + 4·83-s + 4·85-s − 6·89-s − 8·95-s + 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.20·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.371·29-s + 1.64·37-s − 0.937·41-s + 0.609·43-s + 0.824·53-s − 1.07·55-s − 0.520·59-s + 0.768·61-s + 1.48·65-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.439·83-s + 0.433·85-s − 0.635·89-s − 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.59825801356115, −14.90976032313710, −14.38009940692559, −13.72923332620193, −13.32679034686994, −13.07653269687194, −12.36072362273642, −11.69623912790644, −11.08752063337224, −10.57158085352897, −10.05842511579625, −9.713495758100589, −8.802877182003785, −8.443200974906607, −7.830586419754088, −7.285145742757054, −6.196397506640142, −6.044460599937452, −5.606001819538911, −4.708803057303018, −4.028808521337184, −3.392288874489316, −2.479917686562370, −1.973120405548112, −1.144793199098371, 0, 1.144793199098371, 1.973120405548112, 2.479917686562370, 3.392288874489316, 4.028808521337184, 4.708803057303018, 5.606001819538911, 6.044460599937452, 6.196397506640142, 7.285145742757054, 7.830586419754088, 8.443200974906607, 8.802877182003785, 9.713495758100589, 10.05842511579625, 10.57158085352897, 11.08752063337224, 11.69623912790644, 12.36072362273642, 13.07653269687194, 13.32679034686994, 13.72923332620193, 14.38009940692559, 14.90976032313710, 15.59825801356115

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