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Properties

Label 2-168e2-1.1-c1-0-141
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  + 4·5-s − 4·11-s − 4·13-s − 4·19-s + 11·25-s + 2·29-s + 8·31-s + 6·37-s − 4·43-s + 8·47-s − 10·53-s − 16·55-s + 4·59-s + 4·61-s − 16·65-s − 4·67-s − 8·71-s − 16·73-s − 8·79-s − 12·83-s − 8·89-s − 16·95-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.20·11-s − 1.10·13-s − 0.917·19-s + 11/5·25-s + 0.371·29-s + 1.43·31-s + 0.986·37-s − 0.609·43-s + 1.16·47-s − 1.37·53-s − 2.15·55-s + 0.520·59-s + 0.512·61-s − 1.98·65-s − 0.488·67-s − 0.949·71-s − 1.87·73-s − 0.900·79-s − 1.31·83-s − 0.847·89-s − 1.64·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.49785627294221, −14.76060330419197, −14.43933327739092, −13.81416516513793, −13.37181995213581, −12.83967910848754, −12.57563950739172, −11.74257000078755, −11.07734232071096, −10.33652385263266, −10.04417326872776, −9.817080663599103, −8.886943476121600, −8.562276894718932, −7.707172887730964, −7.184670692286781, −6.377808930075782, −6.020267343068048, −5.377058845643518, −4.804771939957339, −4.326014240734032, −2.910148546152156, −2.651817932690078, −2.034999899638850, −1.183404037868524, 0, 1.183404037868524, 2.034999899638850, 2.651817932690078, 2.910148546152156, 4.326014240734032, 4.804771939957339, 5.377058845643518, 6.020267343068048, 6.377808930075782, 7.184670692286781, 7.707172887730964, 8.562276894718932, 8.886943476121600, 9.817080663599103, 10.04417326872776, 10.33652385263266, 11.07734232071096, 11.74257000078755, 12.57563950739172, 12.83967910848754, 13.37181995213581, 13.81416516513793, 14.43933327739092, 14.76060330419197, 15.49785627294221

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