Properties

Label 2-168e2-1.1-c1-0-119
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 + 4·17-s + 6·19-s − 4·23-s − 25-s − 6·29-s + 4·31-s + 6·37-s − 4·41-s + 12·43-s − 12·47-s + 6·53-s − 8·55-s − 6·59-s + 6·61-s − 12·65-s + 12·67-s + 8·71-s − 6·83-s + 8·85-s + 16·89-s + 12·95-s − 12·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.20·11-s − 1.66·13-s + 0.970·17-s + 1.37·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s − 0.624·41-s + 1.82·43-s − 1.75·47-s + 0.824·53-s − 1.07·55-s − 0.781·59-s + 0.768·61-s − 1.48·65-s + 1.46·67-s + 0.949·71-s − 0.658·83-s + 0.867·85-s + 1.69·89-s + 1.23·95-s − 1.21·97-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: $\chi_{28224} (1, \cdot )$
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 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−15.52404195282142, −14.86361004639690, −14.27159449574615, −14.03767130633125, −13.30152691879750, −12.89908260900750, −12.31187467101445, −11.78972585784141, −11.24634478706192, −10.38586591493197, −9.989830464319888, −9.628002670252124, −9.253858215026559, −8.076796642283467, −7.812962258815990, −7.349142163148044, −6.553189465683261, −5.779491183218510, −5.326954135036494, −5.010984132039218, −4.067496984506897, −3.207870175588840, −2.517500753178658, −2.093342995697484, −1.051836562559962, 0, 1.051836562559962, 2.093342995697484, 2.517500753178658, 3.207870175588840, 4.067496984506897, 5.010984132039218, 5.326954135036494, 5.779491183218510, 6.553189465683261, 7.349142163148044, 7.812962258815990, 8.076796642283467, 9.253858215026559, 9.628002670252124, 9.989830464319888, 10.38586591493197, 11.24634478706192, 11.78972585784141, 12.31187467101445, 12.89908260900750, 13.30152691879750, 14.03767130633125, 14.27159449574615, 14.86361004639690, 15.52404195282142

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