Properties

Label 2-92400-1.1-c1-0-112
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 − 27-s − 10·29-s + 8·31-s − 33-s − 6·37-s + 6·39-s + 10·41-s + 4·43-s + 8·47-s + 49-s + 2·51-s − 6·53-s − 4·57-s + 12·59-s + 6·61-s + 63-s − 4·67-s − 16·71-s − 10·73-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 − 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.174·33-s − 0.986·37-s + 0.960·39-s + 1.56·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s + 0.768·61-s + 0.125·63-s − 0.488·67-s − 1.89·71-s − 1.17·73-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: $\chi_{92400} (1, \cdot )$
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 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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

−14.15065013347196, −13.58131517082312, −13.03952859553689, −12.51760981535623, −12.04832847212552, −11.67409703066253, −11.21859113133977, −10.68858729925427, −10.03255194023362, −9.769109268974481, −9.079667728728619, −8.770576580521007, −7.764423754551926, −7.537100559157722, −7.085482469301113, −6.481920030242326, −5.712844577792787, −5.441090454101243, −4.768198089805954, −4.316980148129040, −3.742183858365665, −2.814544844930054, −2.350297887331330, −1.596051999477958, −0.8105763018600230, 0, 0.8105763018600230, 1.596051999477958, 2.350297887331330, 2.814544844930054, 3.742183858365665, 4.316980148129040, 4.768198089805954, 5.441090454101243, 5.712844577792787, 6.481920030242326, 7.085482469301113, 7.537100559157722, 7.764423754551926, 8.770576580521007, 9.079667728728619, 9.769109268974481, 10.03255194023362, 10.68858729925427, 11.21859113133977, 11.67409703066253, 12.04832847212552, 12.51760981535623, 13.03952859553689, 13.58131517082312, 14.15065013347196

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