Properties

Label 2-2760-1.1-c1-0-13
Degree $2$
Conductor $2760$
Sign $1$
Analytic cond. $22.0387$
Root an. cond. $4.69454$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 2.62·7-s + 9-s − 6.55·11-s + 7.06·13-s − 15-s + 6.42·17-s − 1.80·19-s − 2.62·21-s − 23-s + 25-s − 27-s − 7.17·29-s + 4.10·31-s + 6.55·33-s + 2.62·35-s + 4.62·37-s − 7.06·39-s + 4.30·41-s + 6.87·43-s + 45-s + 1.80·47-s − 0.132·49-s − 6.42·51-s − 4.93·53-s − 6.55·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.990·7-s + 0.333·9-s − 1.97·11-s + 1.96·13-s − 0.258·15-s + 1.55·17-s − 0.413·19-s − 0.571·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s − 1.33·29-s + 0.737·31-s + 1.14·33-s + 0.442·35-s + 0.759·37-s − 1.13·39-s + 0.672·41-s + 1.04·43-s + 0.149·45-s + 0.263·47-s − 0.0189·49-s − 0.899·51-s − 0.678·53-s − 0.884·55-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.874445560\)
\(L(\frac12)\) \(\approx\) \(1.874445560\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.62T + 7T^{2} \)
11 \( 1 + 6.55T + 11T^{2} \)
13 \( 1 - 7.06T + 13T^{2} \)
17 \( 1 - 6.42T + 17T^{2} \)
19 \( 1 + 1.80T + 19T^{2} \)
29 \( 1 + 7.17T + 29T^{2} \)
31 \( 1 - 4.10T + 31T^{2} \)
37 \( 1 - 4.62T + 37T^{2} \)
41 \( 1 - 4.30T + 41T^{2} \)
43 \( 1 - 6.87T + 43T^{2} \)
47 \( 1 - 1.80T + 47T^{2} \)
53 \( 1 + 4.93T + 53T^{2} \)
59 \( 1 + 7.54T + 59T^{2} \)
61 \( 1 - 2.70T + 61T^{2} \)
67 \( 1 - 7.37T + 67T^{2} \)
71 \( 1 - 3.93T + 71T^{2} \)
73 \( 1 + 5.04T + 73T^{2} \)
79 \( 1 - 6.11T + 79T^{2} \)
83 \( 1 + 8.42T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 17.9T + 97T^{2} \)
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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

−8.663328678828738534387291644166, −7.912008957227777081733627541938, −7.59309669667721712319670299883, −6.21836841087234510381836919492, −5.70397486593223759304657266723, −5.14443755231330115786808337543, −4.17827882382927729522881134874, −3.10036328468574466465834597737, −1.94773342832698858625827619317, −0.919255680418646679446121166177, 0.919255680418646679446121166177, 1.94773342832698858625827619317, 3.10036328468574466465834597737, 4.17827882382927729522881134874, 5.14443755231330115786808337543, 5.70397486593223759304657266723, 6.21836841087234510381836919492, 7.59309669667721712319670299883, 7.912008957227777081733627541938, 8.663328678828738534387291644166

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