Properties

Label 2-44400-1.1-c1-0-36
Degree $2$
Conductor $44400$
Sign $-1$
Analytic cond. $354.535$
Root an. cond. $18.8291$
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  − 3-s + 9-s − 6·13-s − 2·17-s + 8·19-s − 27-s − 2·29-s − 8·31-s − 37-s + 6·39-s + 10·41-s + 4·43-s + 4·47-s − 7·49-s + 2·51-s − 6·53-s − 8·57-s + 4·59-s + 2·61-s − 12·67-s + 8·71-s − 2·73-s − 8·79-s + 81-s − 4·83-s + 2·87-s + 10·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.66·13-s − 0.485·17-s + 1.83·19-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.164·37-s + 0.960·39-s + 1.56·41-s + 0.609·43-s + 0.583·47-s − 49-s + 0.280·51-s − 0.824·53-s − 1.05·57-s + 0.520·59-s + 0.256·61-s − 1.46·67-s + 0.949·71-s − 0.234·73-s − 0.900·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s + 1.05·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(354.535\)
Root analytic conductor: \(18.8291\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 44400,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
37 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.81548236176068, −14.37217767445982, −14.07316043648524, −13.18199568186918, −12.85021918269737, −12.27766505328285, −11.84258628117376, −11.29050713090756, −10.89032376752245, −10.16211886995179, −9.661137429232537, −9.312273365007525, −8.720772062992455, −7.649505474747891, −7.482209619276428, −7.085593754270570, −6.181082092222408, −5.704934693543355, −5.049920438869072, −4.722113842996370, −3.908816881971299, −3.196519146746966, −2.481293849551964, −1.783053602967056, −0.8459205207675514, 0, 0.8459205207675514, 1.783053602967056, 2.481293849551964, 3.196519146746966, 3.908816881971299, 4.722113842996370, 5.049920438869072, 5.704934693543355, 6.181082092222408, 7.085593754270570, 7.482209619276428, 7.649505474747891, 8.720772062992455, 9.312273365007525, 9.661137429232537, 10.16211886995179, 10.89032376752245, 11.29050713090756, 11.84258628117376, 12.27766505328285, 12.85021918269737, 13.18199568186918, 14.07316043648524, 14.37217767445982, 14.81548236176068

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