Properties

Label 2-1088-1.1-c3-0-14
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  + 3.18·3-s − 14.3·5-s − 17.1·7-s − 16.8·9-s − 3.82·11-s − 4.03·13-s − 45.5·15-s + 17·17-s + 36.1·19-s − 54.5·21-s − 161.·23-s + 80.1·25-s − 139.·27-s + 94.5·29-s − 40.4·31-s − 12.1·33-s + 245.·35-s − 9.76·37-s − 12.8·39-s − 387.·41-s + 169.·43-s + 241.·45-s − 284.·47-s − 49.1·49-s + 54.0·51-s + 602.·53-s + 54.7·55-s + ⋯
L(s)  = 1  + 0.612·3-s − 1.28·5-s − 0.925·7-s − 0.625·9-s − 0.104·11-s − 0.0861·13-s − 0.784·15-s + 0.242·17-s + 0.436·19-s − 0.566·21-s − 1.46·23-s + 0.641·25-s − 0.994·27-s + 0.605·29-s − 0.234·31-s − 0.0641·33-s + 1.18·35-s − 0.0434·37-s − 0.0527·39-s − 1.47·41-s + 0.602·43-s + 0.800·45-s − 0.883·47-s − 0.143·49-s + 0.148·51-s + 1.56·53-s + 0.134·55-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

\(L(2)\) \(\approx\) \(0.9568539228\)
\(L(\frac12)\) \(\approx\) \(0.9568539228\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.18T + 27T^{2} \)
5 \( 1 + 14.3T + 125T^{2} \)
7 \( 1 + 17.1T + 343T^{2} \)
11 \( 1 + 3.82T + 1.33e3T^{2} \)
13 \( 1 + 4.03T + 2.19e3T^{2} \)
19 \( 1 - 36.1T + 6.85e3T^{2} \)
23 \( 1 + 161.T + 1.21e4T^{2} \)
29 \( 1 - 94.5T + 2.43e4T^{2} \)
31 \( 1 + 40.4T + 2.97e4T^{2} \)
37 \( 1 + 9.76T + 5.06e4T^{2} \)
41 \( 1 + 387.T + 6.89e4T^{2} \)
43 \( 1 - 169.T + 7.95e4T^{2} \)
47 \( 1 + 284.T + 1.03e5T^{2} \)
53 \( 1 - 602.T + 1.48e5T^{2} \)
59 \( 1 - 16.2T + 2.05e5T^{2} \)
61 \( 1 - 784.T + 2.26e5T^{2} \)
67 \( 1 + 270.T + 3.00e5T^{2} \)
71 \( 1 - 936.T + 3.57e5T^{2} \)
73 \( 1 - 470.T + 3.89e5T^{2} \)
79 \( 1 - 1.37e3T + 4.93e5T^{2} \)
83 \( 1 - 601.T + 5.71e5T^{2} \)
89 \( 1 - 117.T + 7.04e5T^{2} \)
97 \( 1 + 279.T + 9.12e5T^{2} \)
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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

−9.477194521524479446299861330330, −8.433005572929364427912428806783, −8.035076601314980965001782932616, −7.13560152957443004413013992496, −6.21358542668734878802660963294, −5.13224700534912523840359676066, −3.81817686736956717698990815839, −3.41112062737171739424299952140, −2.30813872066206093407203356991, −0.46963933834069887154160048499, 0.46963933834069887154160048499, 2.30813872066206093407203356991, 3.41112062737171739424299952140, 3.81817686736956717698990815839, 5.13224700534912523840359676066, 6.21358542668734878802660963294, 7.13560152957443004413013992496, 8.035076601314980965001782932616, 8.433005572929364427912428806783, 9.477194521524479446299861330330

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