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Properties

Label 2-92400-1.1-c1-0-109
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 − 3·17-s + 3·19-s − 21-s − 23-s − 27-s − 7·29-s − 6·31-s + 33-s + 8·37-s − 2·41-s − 5·43-s − 6·47-s + 49-s + 3·51-s + 3·53-s − 3·57-s + 5·59-s − 7·61-s + 63-s + 6·67-s + 69-s + 10·71-s − 12·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.727·17-s + 0.688·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s − 1.29·29-s − 1.07·31-s + 0.174·33-s + 1.31·37-s − 0.312·41-s − 0.762·43-s − 0.875·47-s + 1/7·49-s + 0.420·51-s + 0.412·53-s − 0.397·57-s + 0.650·59-s − 0.896·61-s + 0.125·63-s + 0.733·67-s + 0.120·69-s + 1.18·71-s − 1.40·73-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: \(1\)
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 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.01076463300152, −13.51528121057411, −13.07234786689018, −12.66291210943949, −12.07109086055896, −11.46791339981927, −11.21000106850233, −10.80386604577532, −10.11870276606852, −9.642006538184578, −9.198465155481313, −8.575823994392125, −7.954418139319111, −7.539031278892018, −6.985640090888644, −6.447389502010990, −5.818186430346925, −5.340481105239293, −4.888017713961826, −4.234415502008933, −3.674629267825101, −3.010745535185545, −2.136305406191786, −1.705519583411861, −0.7912134512306486, 0, 0.7912134512306486, 1.705519583411861, 2.136305406191786, 3.010745535185545, 3.674629267825101, 4.234415502008933, 4.888017713961826, 5.340481105239293, 5.818186430346925, 6.447389502010990, 6.985640090888644, 7.539031278892018, 7.954418139319111, 8.575823994392125, 9.198465155481313, 9.642006538184578, 10.11870276606852, 10.80386604577532, 11.21000106850233, 11.46791339981927, 12.07109086055896, 12.66291210943949, 13.07234786689018, 13.51528121057411, 14.01076463300152

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