Properties

Label 2-92400-1.1-c1-0-100
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 − 2·17-s + 6·19-s + 21-s − 27-s − 8·31-s + 33-s + 2·37-s + 2·39-s − 4·41-s − 4·43-s − 2·47-s + 49-s + 2·51-s + 6·53-s − 6·57-s + 14·61-s − 63-s − 2·67-s − 4·71-s + 10·73-s + 77-s − 10·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.485·17-s + 1.37·19-s + 0.218·21-s − 0.192·27-s − 1.43·31-s + 0.174·33-s + 0.328·37-s + 0.320·39-s − 0.624·41-s − 0.609·43-s − 0.291·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.794·57-s + 1.79·61-s − 0.125·63-s − 0.244·67-s − 0.474·71-s + 1.17·73-s + 0.113·77-s − 1.12·79-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: $\chi_{92400} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.09999459781519, −13.47026024004818, −13.01175076254762, −12.72369805648172, −12.00376032322495, −11.63998580653213, −11.24601327791342, −10.60477311840591, −10.11095262470552, −9.691504912922509, −9.227702034231353, −8.600842610698557, −8.022733183462411, −7.279204549591584, −7.121844736420420, −6.486143288250404, −5.786321040696985, −5.344013248646998, −4.951328241473008, −4.204350839301092, −3.599922196431781, −3.013296272937875, −2.295385798911168, −1.610334164128601, −0.7480205398302898, 0, 0.7480205398302898, 1.610334164128601, 2.295385798911168, 3.013296272937875, 3.599922196431781, 4.204350839301092, 4.951328241473008, 5.344013248646998, 5.786321040696985, 6.486143288250404, 7.121844736420420, 7.279204549591584, 8.022733183462411, 8.600842610698557, 9.227702034231353, 9.691504912922509, 10.11095262470552, 10.60477311840591, 11.24601327791342, 11.63998580653213, 12.00376032322495, 12.72369805648172, 13.01175076254762, 13.47026024004818, 14.09999459781519

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