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Properties

Label 2-92400-1.1-c1-0-105
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s + 11-s + 4·13-s + 6·17-s − 4·19-s − 21-s + 6·23-s + 27-s + 6·31-s + 33-s + 8·37-s + 4·39-s − 6·41-s + 4·43-s + 8·47-s + 49-s + 6·51-s − 6·53-s − 4·57-s + 4·59-s + 6·61-s − 63-s + 12·67-s + 6·69-s − 4·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 1.45·17-s − 0.917·19-s − 0.218·21-s + 1.25·23-s + 0.192·27-s + 1.07·31-s + 0.174·33-s + 1.31·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s + 0.768·61-s − 0.125·63-s + 1.46·67-s + 0.722·69-s − 0.468·73-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: \(0\)
Selberg data: \((2,\ 92400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.515079643\)
\(L(\frac12)\) \(\approx\) \(4.515079643\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−13.75699373688945, −13.39028137043242, −12.89811859245177, −12.50289275342715, −11.89958408622708, −11.38797359476617, −10.78637486284465, −10.39934702513464, −9.763091577357899, −9.390282058303831, −8.795005189873860, −8.324889352317616, −7.954986756986705, −7.243630890716487, −6.747607285415614, −6.194516980037825, −5.732834601232275, −5.030846195563652, −4.344762604817949, −3.813054137126217, −3.272401936595706, −2.766429338953785, −2.042190065045576, −1.142680555249196, −0.7622951911478453, 0.7622951911478453, 1.142680555249196, 2.042190065045576, 2.766429338953785, 3.272401936595706, 3.813054137126217, 4.344762604817949, 5.030846195563652, 5.732834601232275, 6.194516980037825, 6.747607285415614, 7.243630890716487, 7.954986756986705, 8.324889352317616, 8.795005189873860, 9.390282058303831, 9.763091577357899, 10.39934702513464, 10.78637486284465, 11.38797359476617, 11.89958408622708, 12.50289275342715, 12.89811859245177, 13.39028137043242, 13.75699373688945

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