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Properties

Label 2-168e2-1.1-c1-0-140
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 − 6·13-s − 8·17-s + 11·25-s + 4·29-s + 2·37-s + 8·41-s + 4·53-s − 10·61-s − 24·65-s − 6·73-s − 32·85-s − 16·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.66·13-s − 1.94·17-s + 11/5·25-s + 0.742·29-s + 0.328·37-s + 1.24·41-s + 0.549·53-s − 1.28·61-s − 2.97·65-s − 0.702·73-s − 3.47·85-s − 1.69·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−15.35152379815575, −14.89631515667483, −14.29124480067684, −13.90607580557439, −13.40013360357369, −12.85002677827371, −12.49979413956820, −11.74578132502001, −11.07109853833458, −10.49861159125948, −10.05956947532612, −9.466927689256943, −9.129755986414599, −8.574598391947072, −7.697843905293990, −6.988154421225719, −6.608250983392829, −5.941340653256668, −5.441094189492060, −4.599775408302182, −4.455960268382456, −3.067456604777924, −2.359791957629303, −2.185365193665095, −1.182043340658777, 0, 1.182043340658777, 2.185365193665095, 2.359791957629303, 3.067456604777924, 4.455960268382456, 4.599775408302182, 5.441094189492060, 5.941340653256668, 6.608250983392829, 6.988154421225719, 7.697843905293990, 8.574598391947072, 9.129755986414599, 9.466927689256943, 10.05956947532612, 10.49861159125948, 11.07109853833458, 11.74578132502001, 12.49979413956820, 12.85002677827371, 13.40013360357369, 13.90607580557439, 14.29124480067684, 14.89631515667483, 15.35152379815575

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