Properties

Label 2-168e2-1.1-c1-0-120
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  + 5-s − 5·11-s + 4·17-s + 8·19-s − 4·23-s − 4·25-s − 5·29-s − 3·31-s + 4·37-s + 2·43-s − 6·47-s − 9·53-s − 5·55-s + 11·59-s + 6·61-s − 2·67-s + 2·71-s + 10·73-s − 3·79-s + 7·83-s + 4·85-s + 6·89-s + 8·95-s + 7·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.50·11-s + 0.970·17-s + 1.83·19-s − 0.834·23-s − 4/5·25-s − 0.928·29-s − 0.538·31-s + 0.657·37-s + 0.304·43-s − 0.875·47-s − 1.23·53-s − 0.674·55-s + 1.43·59-s + 0.768·61-s − 0.244·67-s + 0.237·71-s + 1.17·73-s − 0.337·79-s + 0.768·83-s + 0.433·85-s + 0.635·89-s + 0.820·95-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−15.63918725501311, −14.84993105217311, −14.42097770333559, −13.79627026085608, −13.42794876509916, −12.84914366827196, −12.38789092656731, −11.60362260264025, −11.32070933917668, −10.51638143503017, −9.939524822074534, −9.709100088474752, −9.087234155869149, −8.141737574762417, −7.699005374037979, −7.495172838002475, −6.497981065982113, −5.818553329463610, −5.308067881858481, −5.038860133552262, −3.891625136276179, −3.374678722907713, −2.610876053246616, −1.957680767301689, −1.055132237949372, 0, 1.055132237949372, 1.957680767301689, 2.610876053246616, 3.374678722907713, 3.891625136276179, 5.038860133552262, 5.308067881858481, 5.818553329463610, 6.497981065982113, 7.495172838002475, 7.699005374037979, 8.141737574762417, 9.087234155869149, 9.709100088474752, 9.939524822074534, 10.51638143503017, 11.32070933917668, 11.60362260264025, 12.38789092656731, 12.84914366827196, 13.42794876509916, 13.79627026085608, 14.42097770333559, 14.84993105217311, 15.63918725501311

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