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Properties

Label 2-92400-1.1-c1-0-167
Degree $2$
Conductor $92400$
Sign $-1$
Analytic cond. $737.817$
Root an. cond. $27.1628$
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 + 7-s + 9-s − 11-s + 6·19-s − 21-s + 2·23-s − 27-s + 2·29-s + 6·31-s + 33-s + 2·37-s + 4·41-s − 11·43-s − 3·47-s + 49-s − 3·53-s − 6·57-s + 14·59-s + 2·61-s + 63-s − 12·67-s − 2·69-s + 4·71-s − 3·73-s − 77-s − 8·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.37·19-s − 0.218·21-s + 0.417·23-s − 0.192·27-s + 0.371·29-s + 1.07·31-s + 0.174·33-s + 0.328·37-s + 0.624·41-s − 1.67·43-s − 0.437·47-s + 1/7·49-s − 0.412·53-s − 0.794·57-s + 1.82·59-s + 0.256·61-s + 0.125·63-s − 1.46·67-s − 0.240·69-s + 0.474·71-s − 0.351·73-s − 0.113·77-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(737.817\)
Root analytic conductor: \(27.1628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−14.07929733424241, −13.47998023710775, −13.16556261625525, −12.60581021417046, −11.92482182855655, −11.64435698083871, −11.29661964687019, −10.61086966599345, −10.12021369052678, −9.764678551647400, −9.172359001264098, −8.451358679912570, −8.115015679338601, −7.440312222356633, −7.019063331643701, −6.453914802352030, −5.819671493542603, −5.330896629616960, −4.828836856129488, −4.372607500049639, −3.563896507726208, −2.976800203034052, −2.354841330440552, −1.412753353882580, −0.9707806514070782, 0, 0.9707806514070782, 1.412753353882580, 2.354841330440552, 2.976800203034052, 3.563896507726208, 4.372607500049639, 4.828836856129488, 5.330896629616960, 5.819671493542603, 6.453914802352030, 7.019063331643701, 7.440312222356633, 8.115015679338601, 8.451358679912570, 9.172359001264098, 9.764678551647400, 10.12021369052678, 10.61086966599345, 11.29661964687019, 11.64435698083871, 11.92482182855655, 12.60581021417046, 13.16556261625525, 13.47998023710775, 14.07929733424241

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