Properties

Label 2-168e2-1.1-c1-0-12
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 $0$

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: $\chi_{28224} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9176612951\)
\(L(\frac12)\) \(\approx\) \(0.9176612951\)
\(L(\frac{3}{2})\) 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

−15.39519542102159, −14.75179490465855, −14.08677123287178, −13.62098727862923, −13.17281830172794, −12.50474883141693, −11.79165081153529, −11.49618746227484, −11.19662668803280, −10.19636996861544, −10.00673375569603, −9.018851955196253, −8.628615603679603, −8.131139028843569, −7.602970715179927, −6.637975250159178, −6.513475164490255, −5.806943550263398, −4.694479634017919, −4.476585257977034, −3.682057261205858, −3.247031677214063, −2.156370387336091, −1.550318801564893, −0.3671528183413931, 0.3671528183413931, 1.550318801564893, 2.156370387336091, 3.247031677214063, 3.682057261205858, 4.476585257977034, 4.694479634017919, 5.806943550263398, 6.513475164490255, 6.637975250159178, 7.602970715179927, 8.131139028843569, 8.628615603679603, 9.018851955196253, 10.00673375569603, 10.19636996861544, 11.19662668803280, 11.49618746227484, 11.79165081153529, 12.50474883141693, 13.17281830172794, 13.62098727862923, 14.08677123287178, 14.75179490465855, 15.39519542102159

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