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Properties

Label 2-92400-1.1-c1-0-128
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 − 2·13-s + 6·17-s + 4·19-s − 21-s − 8·23-s − 27-s − 6·29-s − 4·31-s − 33-s − 2·37-s + 2·39-s − 6·41-s + 12·43-s + 49-s − 6·51-s − 2·53-s − 4·57-s − 4·59-s − 2·61-s + 63-s − 8·67-s + 8·69-s + 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.218·21-s − 1.66·23-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s − 0.328·37-s + 0.320·39-s − 0.937·41-s + 1.82·43-s + 1/7·49-s − 0.840·51-s − 0.274·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s − 0.977·67-s + 0.963·69-s + 0.949·71-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.10588213954427, −13.76352953751660, −12.94452917780800, −12.47776323688771, −12.13518993051633, −11.57095953865393, −11.35947052423880, −10.50974643579139, −10.19638824671843, −9.684656191973191, −9.217470823222170, −8.605814788432109, −7.825572074207189, −7.481307102124863, −7.244441441221975, −6.217274716625980, −5.903369296637758, −5.388487650038409, −4.891427238657851, −4.195380772086232, −3.630174670561597, −3.101438921010251, −2.125290753657835, −1.629364611345321, −0.8671218792328161, 0, 0.8671218792328161, 1.629364611345321, 2.125290753657835, 3.101438921010251, 3.630174670561597, 4.195380772086232, 4.891427238657851, 5.388487650038409, 5.903369296637758, 6.217274716625980, 7.244441441221975, 7.481307102124863, 7.825572074207189, 8.605814788432109, 9.217470823222170, 9.684656191973191, 10.19638824671843, 10.50974643579139, 11.35947052423880, 11.57095953865393, 12.13518993051633, 12.47776323688771, 12.94452917780800, 13.76352953751660, 14.10588213954427

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