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Properties

Label 2-32340-1.1-c1-0-26
Degree $2$
Conductor $32340$
Sign $-1$
Analytic cond. $258.236$
Root an. cond. $16.0697$
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 + 5-s + 9-s + 11-s − 2·13-s − 15-s − 2·19-s + 25-s − 27-s − 8·31-s − 33-s + 2·37-s + 2·39-s + 2·43-s + 45-s + 6·53-s + 55-s + 2·57-s + 12·59-s − 2·61-s − 2·65-s − 4·67-s − 2·73-s − 75-s − 10·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.258·15-s − 0.458·19-s + 1/5·25-s − 0.192·27-s − 1.43·31-s − 0.174·33-s + 0.328·37-s + 0.320·39-s + 0.304·43-s + 0.149·45-s + 0.824·53-s + 0.134·55-s + 0.264·57-s + 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s − 0.234·73-s − 0.115·75-s − 1.12·79-s + 1/9·81-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32340\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(258.236\)
Root analytic conductor: \(16.0697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.19485465374939, −14.71330361086253, −14.43769125426639, −13.58434527260404, −13.20955793705807, −12.63645117089189, −12.13465555918353, −11.64695516704369, −10.96075853638642, −10.61677239814308, −9.939059926386121, −9.482749719995483, −8.919718048398488, −8.301459121607547, −7.537935320157306, −7.015331118839470, −6.532802288428873, −5.730523904397679, −5.472874009320031, −4.661112492574878, −4.106777834789862, −3.372184619488838, −2.459897329292420, −1.864739138484910, −0.9808778058578225, 0, 0.9808778058578225, 1.864739138484910, 2.459897329292420, 3.372184619488838, 4.106777834789862, 4.661112492574878, 5.472874009320031, 5.730523904397679, 6.532802288428873, 7.015331118839470, 7.537935320157306, 8.301459121607547, 8.919718048398488, 9.482749719995483, 9.939059926386121, 10.61677239814308, 10.96075853638642, 11.64695516704369, 12.13465555918353, 12.63645117089189, 13.20955793705807, 13.58434527260404, 14.43769125426639, 14.71330361086253, 15.19485465374939

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