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Properties

Label 2-168e2-1.1-c1-0-151
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  + 3·5-s − 11-s + 4·13-s − 4·17-s − 8·23-s + 4·25-s − 7·29-s + 11·31-s − 4·37-s + 4·41-s + 2·43-s + 2·47-s − 11·53-s − 3·55-s + 7·59-s − 10·61-s + 12·65-s − 10·67-s − 6·71-s − 6·73-s + 11·79-s + 11·83-s − 12·85-s − 6·89-s + 7·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.301·11-s + 1.10·13-s − 0.970·17-s − 1.66·23-s + 4/5·25-s − 1.29·29-s + 1.97·31-s − 0.657·37-s + 0.624·41-s + 0.304·43-s + 0.291·47-s − 1.51·53-s − 0.404·55-s + 0.911·59-s − 1.28·61-s + 1.48·65-s − 1.22·67-s − 0.712·71-s − 0.702·73-s + 1.23·79-s + 1.20·83-s − 1.30·85-s − 0.635·89-s + 0.710·97-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: Trivial
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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−15.73291831064270, −14.87627170516456, −14.30223834239594, −13.69834465215811, −13.44762846971475, −13.11236309973230, −12.22544440668939, −11.83514982254222, −10.99175903198080, −10.61218956751218, −10.13232638457968, −9.329100496116177, −9.235747175970351, −8.277843312045185, −7.987234371114598, −7.057138228000399, −6.356458627424024, −6.012004355581490, −5.580176700396405, −4.675945295128332, −4.143158172915856, −3.305820030319768, −2.476859348686605, −1.917858718663878, −1.244227301367295, 0, 1.244227301367295, 1.917858718663878, 2.476859348686605, 3.305820030319768, 4.143158172915856, 4.675945295128332, 5.580176700396405, 6.012004355581490, 6.356458627424024, 7.057138228000399, 7.987234371114598, 8.277843312045185, 9.235747175970351, 9.329100496116177, 10.13232638457968, 10.61218956751218, 10.99175903198080, 11.83514982254222, 12.22544440668939, 13.11236309973230, 13.44762846971475, 13.69834465215811, 14.30223834239594, 14.87627170516456, 15.73291831064270

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