Properties

Label 2-1088-1.1-c3-0-41
Degree $2$
Conductor $1088$
Sign $1$
Analytic cond. $64.1940$
Root an. cond. $8.01212$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.76·3-s − 3.36·5-s + 10.1·7-s + 6.21·9-s − 19.6·11-s + 68.5·13-s − 19.3·15-s + 17·17-s + 28.5·19-s + 58.6·21-s + 201.·23-s − 113.·25-s − 119.·27-s − 158.·29-s + 72.2·31-s − 113.·33-s − 34.2·35-s + 109.·37-s + 395.·39-s + 12.8·41-s + 431.·43-s − 20.9·45-s − 119.·47-s − 239.·49-s + 97.9·51-s + 341.·53-s + 66.1·55-s + ⋯
L(s)  = 1  + 1.10·3-s − 0.300·5-s + 0.549·7-s + 0.230·9-s − 0.538·11-s + 1.46·13-s − 0.333·15-s + 0.242·17-s + 0.344·19-s + 0.609·21-s + 1.82·23-s − 0.909·25-s − 0.853·27-s − 1.01·29-s + 0.418·31-s − 0.597·33-s − 0.165·35-s + 0.484·37-s + 1.62·39-s + 0.0487·41-s + 1.52·43-s − 0.0692·45-s − 0.370·47-s − 0.697·49-s + 0.269·51-s + 0.883·53-s + 0.162·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $1$
Analytic conductor: \(64.1940\)
Root analytic conductor: \(8.01212\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.387216705\)
\(L(\frac12)\) \(\approx\) \(3.387216705\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 - 17T \)
good3 \( 1 - 5.76T + 27T^{2} \)
5 \( 1 + 3.36T + 125T^{2} \)
7 \( 1 - 10.1T + 343T^{2} \)
11 \( 1 + 19.6T + 1.33e3T^{2} \)
13 \( 1 - 68.5T + 2.19e3T^{2} \)
19 \( 1 - 28.5T + 6.85e3T^{2} \)
23 \( 1 - 201.T + 1.21e4T^{2} \)
29 \( 1 + 158.T + 2.43e4T^{2} \)
31 \( 1 - 72.2T + 2.97e4T^{2} \)
37 \( 1 - 109.T + 5.06e4T^{2} \)
41 \( 1 - 12.8T + 6.89e4T^{2} \)
43 \( 1 - 431.T + 7.95e4T^{2} \)
47 \( 1 + 119.T + 1.03e5T^{2} \)
53 \( 1 - 341.T + 1.48e5T^{2} \)
59 \( 1 - 390.T + 2.05e5T^{2} \)
61 \( 1 - 700.T + 2.26e5T^{2} \)
67 \( 1 + 113.T + 3.00e5T^{2} \)
71 \( 1 + 273.T + 3.57e5T^{2} \)
73 \( 1 - 229.T + 3.89e5T^{2} \)
79 \( 1 + 628.T + 4.93e5T^{2} \)
83 \( 1 - 853.T + 5.71e5T^{2} \)
89 \( 1 - 529.T + 7.04e5T^{2} \)
97 \( 1 - 1.23e3T + 9.12e5T^{2} \)
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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

−9.189826700087555554745265596099, −8.681654664339941154221447357248, −7.906301900740961243838244714485, −7.35423346816679088879775247131, −6.06548612732382181212924471283, −5.15307713860644721869233320713, −3.94504663134218658182130861356, −3.23497261456955589148699628900, −2.19157819945870172262358627528, −0.950353427285395608155210991364, 0.950353427285395608155210991364, 2.19157819945870172262358627528, 3.23497261456955589148699628900, 3.94504663134218658182130861356, 5.15307713860644721869233320713, 6.06548612732382181212924471283, 7.35423346816679088879775247131, 7.906301900740961243838244714485, 8.681654664339941154221447357248, 9.189826700087555554745265596099

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