Properties

Label 2-2760-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 3·7-s + 9-s − 2·11-s + 2·13-s + 15-s − 17-s + 6·19-s − 3·21-s − 23-s + 25-s − 27-s + 3·29-s − 3·31-s + 2·33-s − 3·35-s + 3·37-s − 2·39-s − 7·41-s − 45-s + 2·49-s + 51-s − 5·53-s + 2·55-s − 6·57-s + 9·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s − 0.242·17-s + 1.37·19-s − 0.654·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s − 0.538·31-s + 0.348·33-s − 0.507·35-s + 0.493·37-s − 0.320·39-s − 1.09·41-s − 0.149·45-s + 2/7·49-s + 0.140·51-s − 0.686·53-s + 0.269·55-s − 0.794·57-s + 1.17·59-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: \(0\)
Selberg data: \((2,\ 2760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.547510434\)
\(L(\frac12)\) \(\approx\) \(1.547510434\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−8.600603493441043108112734992629, −8.051678688285955756359172584900, −7.37287096715256720923674298587, −6.57782477277849510090399409127, −5.50261515686804496666550153026, −5.05400746324977715175317429041, −4.18587159900465403118560200563, −3.21608552083442073302077741898, −1.93453494393080793716491508792, −0.822655303884078211967146289953, 0.822655303884078211967146289953, 1.93453494393080793716491508792, 3.21608552083442073302077741898, 4.18587159900465403118560200563, 5.05400746324977715175317429041, 5.50261515686804496666550153026, 6.57782477277849510090399409127, 7.37287096715256720923674298587, 8.051678688285955756359172584900, 8.600603493441043108112734992629

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