Properties

Label 2-2760-1.1-c1-0-38
Degree $2$
Conductor $2760$
Sign $-1$
Analytic cond. $22.0387$
Root an. cond. $4.69454$
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 − 2·13-s − 15-s − 6·17-s − 23-s + 25-s + 27-s + 6·29-s − 6·37-s − 2·39-s − 6·41-s − 45-s − 8·47-s − 7·49-s − 6·51-s − 6·53-s − 12·59-s + 10·61-s + 2·65-s − 69-s − 6·73-s + 75-s + 8·79-s + 81-s + 6·85-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.149·45-s − 1.16·47-s − 49-s − 0.840·51-s − 0.824·53-s − 1.56·59-s + 1.28·61-s + 0.248·65-s − 0.120·69-s − 0.702·73-s + 0.115·75-s + 0.900·79-s + 1/9·81-s + 0.650·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(22.0387\)
Root analytic conductor: \(4.69454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2760,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
23 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.405062090501522425432625806457, −7.83315783909893742198109494470, −6.87953484753163495120463930655, −6.42745323646683849474829733946, −5.06850952503014453195679085382, −4.50421650696143290281208242635, −3.54023389135049481483602881039, −2.67869569929630113057820760890, −1.67449794423814169224104342467, 0, 1.67449794423814169224104342467, 2.67869569929630113057820760890, 3.54023389135049481483602881039, 4.50421650696143290281208242635, 5.06850952503014453195679085382, 6.42745323646683849474829733946, 6.87953484753163495120463930655, 7.83315783909893742198109494470, 8.405062090501522425432625806457

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