Properties

Label 2-92400-1.1-c1-0-140
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 − 6·13-s + 2·17-s + 4·19-s + 21-s − 8·23-s + 27-s − 6·29-s − 33-s + 2·37-s − 6·39-s − 10·41-s + 4·43-s + 49-s + 2·51-s + 14·53-s + 4·57-s + 8·59-s + 2·61-s + 63-s − 8·69-s − 8·71-s + 6·73-s − 77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s + 0.218·21-s − 1.66·23-s + 0.192·27-s − 1.11·29-s − 0.174·33-s + 0.328·37-s − 0.960·39-s − 1.56·41-s + 0.609·43-s + 1/7·49-s + 0.280·51-s + 1.92·53-s + 0.529·57-s + 1.04·59-s + 0.256·61-s + 0.125·63-s − 0.963·69-s − 0.949·71-s + 0.702·73-s − 0.113·77-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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.19238317091190, −13.48782463817034, −13.36180486106554, −12.50733615882312, −12.02065611378862, −11.88755015221259, −11.17668807907474, −10.41881958968805, −10.06983149541500, −9.636969927852678, −9.230729232760300, −8.440912647692143, −8.057750576287962, −7.509473412563943, −7.228048455600332, −6.602431209237906, −5.568083774019320, −5.508543418036818, −4.756453660870717, −4.140730897718063, −3.591563006628835, −2.925033817282107, −2.209264295585594, −1.919288264067369, −0.8941768093572040, 0, 0.8941768093572040, 1.919288264067369, 2.209264295585594, 2.925033817282107, 3.591563006628835, 4.140730897718063, 4.756453660870717, 5.508543418036818, 5.568083774019320, 6.602431209237906, 7.228048455600332, 7.509473412563943, 8.057750576287962, 8.440912647692143, 9.230729232760300, 9.636969927852678, 10.06983149541500, 10.41881958968805, 11.17668807907474, 11.88755015221259, 12.02065611378862, 12.50733615882312, 13.36180486106554, 13.48782463817034, 14.19238317091190

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