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Properties

Label 2-92400-1.1-c1-0-127
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 − 4·13-s + 5·17-s + 5·19-s − 21-s − 3·23-s − 27-s − 29-s + 2·31-s + 33-s + 4·37-s + 4·39-s − 10·41-s − 9·43-s + 6·47-s + 49-s − 5·51-s + 7·53-s − 5·57-s + 59-s + 5·61-s + 63-s + 2·67-s + 3·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 1.21·17-s + 1.14·19-s − 0.218·21-s − 0.625·23-s − 0.192·27-s − 0.185·29-s + 0.359·31-s + 0.174·33-s + 0.657·37-s + 0.640·39-s − 1.56·41-s − 1.37·43-s + 0.875·47-s + 1/7·49-s − 0.700·51-s + 0.961·53-s − 0.662·57-s + 0.130·59-s + 0.640·61-s + 0.125·63-s + 0.244·67-s + 0.361·69-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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.98815045604519, −13.69241602054363, −13.06686912852702, −12.43917866787656, −12.06387727189341, −11.68435833858109, −11.32185718966802, −10.48612225594175, −10.10055211117766, −9.816302469053328, −9.250523997320728, −8.454890683927461, −7.944431978603965, −7.574995287869810, −6.989023114689884, −6.528520317792171, −5.680355355193721, −5.297977798691820, −5.029380435130394, −4.234589065835451, −3.634045629745619, −2.954207113770044, −2.312081981684408, −1.533189055075739, −0.8670712241120658, 0, 0.8670712241120658, 1.533189055075739, 2.312081981684408, 2.954207113770044, 3.634045629745619, 4.234589065835451, 5.029380435130394, 5.297977798691820, 5.680355355193721, 6.528520317792171, 6.989023114689884, 7.574995287869810, 7.944431978603965, 8.454890683927461, 9.250523997320728, 9.816302469053328, 10.10055211117766, 10.48612225594175, 11.32185718966802, 11.68435833858109, 12.06387727189341, 12.43917866787656, 13.06686912852702, 13.69241602054363, 13.98815045604519

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