Properties

Label 2-9280-1.1-c1-0-64
Degree $2$
Conductor $9280$
Sign $1$
Analytic cond. $74.1011$
Root an. cond. $8.60820$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 3·9-s + 2·11-s + 2·13-s + 2·19-s − 8·23-s + 25-s − 29-s + 2·31-s + 4·37-s − 10·41-s − 4·43-s − 3·45-s + 12·47-s − 7·49-s + 6·53-s + 2·55-s + 12·59-s + 10·61-s + 2·65-s − 12·67-s + 12·71-s + 12·73-s + 2·79-s + 9·81-s + 4·83-s − 10·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 9-s + 0.603·11-s + 0.554·13-s + 0.458·19-s − 1.66·23-s + 1/5·25-s − 0.185·29-s + 0.359·31-s + 0.657·37-s − 1.56·41-s − 0.609·43-s − 0.447·45-s + 1.75·47-s − 49-s + 0.824·53-s + 0.269·55-s + 1.56·59-s + 1.28·61-s + 0.248·65-s − 1.46·67-s + 1.42·71-s + 1.40·73-s + 0.225·79-s + 81-s + 0.439·83-s − 1.05·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9280\)    =    \(2^{6} \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(74.1011\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.036196340\)
\(L(\frac12)\) \(\approx\) \(2.036196340\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
29 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.87390130263499594655803792880, −6.85878127274005876305289452726, −6.32910911654634090692679217003, −5.67547893331823366119427670837, −5.14702634082636994083914885066, −4.06166120471276212238647497880, −3.51660391035210261528188618620, −2.55648423175736375343924360104, −1.80127533522612248784592622087, −0.68269192299554687018891347858, 0.68269192299554687018891347858, 1.80127533522612248784592622087, 2.55648423175736375343924360104, 3.51660391035210261528188618620, 4.06166120471276212238647497880, 5.14702634082636994083914885066, 5.67547893331823366119427670837, 6.32910911654634090692679217003, 6.85878127274005876305289452726, 7.87390130263499594655803792880

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