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Properties

Label 2-168e2-1.1-c1-0-149
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 + 2·11-s − 3·13-s + 8·17-s − 19-s − 8·23-s − 25-s + 4·29-s − 3·31-s + 37-s + 6·41-s − 11·43-s + 6·47-s − 12·53-s + 4·55-s − 4·59-s − 6·61-s − 6·65-s − 13·67-s + 10·71-s + 11·73-s − 3·79-s − 2·83-s + 16·85-s − 2·95-s − 10·97-s + 101-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.603·11-s − 0.832·13-s + 1.94·17-s − 0.229·19-s − 1.66·23-s − 1/5·25-s + 0.742·29-s − 0.538·31-s + 0.164·37-s + 0.937·41-s − 1.67·43-s + 0.875·47-s − 1.64·53-s + 0.539·55-s − 0.520·59-s − 0.768·61-s − 0.744·65-s − 1.58·67-s + 1.18·71-s + 1.28·73-s − 0.337·79-s − 0.219·83-s + 1.73·85-s − 0.205·95-s − 1.01·97-s + 0.0995·101-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.37622945883544, −14.85600627406414, −14.20871376651419, −14.05897798848052, −13.54911118130582, −12.62573338211791, −12.29626757134310, −11.94340022347161, −11.18574518747935, −10.45979406667389, −9.914900869518971, −9.690232160050121, −9.135861101950606, −8.234032704744059, −7.830824579083802, −7.245514843076347, −6.379745900711446, −6.021335986143494, −5.435620123686271, −4.795367888915619, −4.023027768466597, −3.350289748053283, −2.593236367995230, −1.819605677841179, −1.229733688443348, 0, 1.229733688443348, 1.819605677841179, 2.593236367995230, 3.350289748053283, 4.023027768466597, 4.795367888915619, 5.435620123686271, 6.021335986143494, 6.379745900711446, 7.245514843076347, 7.830824579083802, 8.234032704744059, 9.135861101950606, 9.690232160050121, 9.914900869518971, 10.45979406667389, 11.18574518747935, 11.94340022347161, 12.29626757134310, 12.62573338211791, 13.54911118130582, 14.05897798848052, 14.20871376651419, 14.85600627406414, 15.37622945883544

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