Properties

Label 2-92400-1.1-c1-0-129
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 − 6·17-s − 21-s − 4·23-s − 27-s + 2·29-s + 33-s + 2·37-s − 6·39-s − 2·41-s − 8·43-s + 12·47-s + 49-s + 6·51-s + 6·53-s − 4·59-s + 2·61-s + 63-s − 12·67-s + 4·69-s + 4·71-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 − 1.45·17-s − 0.218·21-s − 0.834·23-s − 0.192·27-s + 0.371·29-s + 0.174·33-s + 0.328·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.520·59-s + 0.256·61-s + 0.125·63-s − 1.46·67-s + 0.481·69-s + 0.474·71-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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.82400790278493, −13.56350995511213, −13.25215210310396, −12.60409373666613, −11.96114348255744, −11.66595721006567, −11.06988045930624, −10.63601280878390, −10.42844347511990, −9.599094659803892, −9.041970649355740, −8.541916131423156, −8.161565406232610, −7.536729726088736, −6.797640516441560, −6.487949154998125, −5.868358933838266, −5.465525528631444, −4.734003122678084, −4.182360693514836, −3.806319120709061, −2.956312945615133, −2.204505669953645, −1.594717354728384, −0.8706356640950264, 0, 0.8706356640950264, 1.594717354728384, 2.204505669953645, 2.956312945615133, 3.806319120709061, 4.182360693514836, 4.734003122678084, 5.465525528631444, 5.868358933838266, 6.487949154998125, 6.797640516441560, 7.536729726088736, 8.161565406232610, 8.541916131423156, 9.041970649355740, 9.599094659803892, 10.42844347511990, 10.63601280878390, 11.06988045930624, 11.66595721006567, 11.96114348255744, 12.60409373666613, 13.25215210310396, 13.56350995511213, 13.82400790278493

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