Properties

Label 2-92400-1.1-c1-0-107
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 − 8·17-s − 8·19-s − 21-s + 6·23-s − 27-s + 2·29-s − 4·31-s − 33-s + 10·41-s + 4·43-s − 4·47-s + 49-s + 8·51-s + 6·53-s + 8·57-s − 6·61-s + 63-s + 2·67-s − 6·69-s + 8·71-s + 2·73-s + 77-s + 12·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.94·17-s − 1.83·19-s − 0.218·21-s + 1.25·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.174·33-s + 1.56·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 1.12·51-s + 0.824·53-s + 1.05·57-s − 0.768·61-s + 0.125·63-s + 0.244·67-s − 0.722·69-s + 0.949·71-s + 0.234·73-s + 0.113·77-s + 1.35·79-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 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−14.07819439219993, −13.47728455972017, −12.98602896434780, −12.70570780718304, −12.16244771523789, −11.47536743054261, −11.00245384321447, −10.82657502687660, −10.39821581327303, −9.408014985828480, −9.169895353258547, −8.635011523023220, −8.129952965999496, −7.432359937753964, −6.811547583220230, −6.517402453411569, −6.022188085514883, −5.243744596800076, −4.791136439536476, −4.141865897381343, −3.952253887402346, −2.782097136056494, −2.290759242312191, −1.660977876516654, −0.7871820561580020, 0, 0.7871820561580020, 1.660977876516654, 2.290759242312191, 2.782097136056494, 3.952253887402346, 4.141865897381343, 4.791136439536476, 5.243744596800076, 6.022188085514883, 6.517402453411569, 6.811547583220230, 7.432359937753964, 8.129952965999496, 8.635011523023220, 9.169895353258547, 9.408014985828480, 10.39821581327303, 10.82657502687660, 11.00245384321447, 11.47536743054261, 12.16244771523789, 12.70570780718304, 12.98602896434780, 13.47728455972017, 14.07819439219993

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