Properties

Label 2-168e2-1.1-c1-0-137
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

−14.99432460024252, −14.90906709999022, −14.66297453019720, −13.90608598700270, −13.29037874056585, −12.93447727722281, −12.14874023900028, −11.92382744839657, −11.21530493039183, −10.46204240464370, −10.04972430964192, −9.498780936250426, −9.091577411363238, −8.505042751369051, −7.634345288740188, −7.154818519318566, −6.612010775366307, −5.911961512995733, −5.429004480770651, −4.766694474789768, −4.055897461988929, −3.397256841728487, −2.443275336972204, −1.978482188568592, −1.163221423632743, 0, 1.163221423632743, 1.978482188568592, 2.443275336972204, 3.397256841728487, 4.055897461988929, 4.766694474789768, 5.429004480770651, 5.911961512995733, 6.612010775366307, 7.154818519318566, 7.634345288740188, 8.505042751369051, 9.091577411363238, 9.498780936250426, 10.04972430964192, 10.46204240464370, 11.21530493039183, 11.92382744839657, 12.14874023900028, 12.93447727722281, 13.29037874056585, 13.90608598700270, 14.66297453019720, 14.90906709999022, 14.99432460024252

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