Properties

Label 2-1088-1.1-c3-0-46
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.2·3-s + 3.08·5-s − 7.31·7-s + 77.6·9-s − 23.9·11-s − 35.5·13-s − 31.5·15-s + 17·17-s + 34.2·19-s + 74.7·21-s − 149.·23-s − 115.·25-s − 517.·27-s + 120.·29-s + 247.·31-s + 244.·33-s − 22.5·35-s + 448.·37-s + 363.·39-s + 303.·41-s + 194.·43-s + 239.·45-s + 21.0·47-s − 289.·49-s − 173.·51-s + 362.·53-s − 73.8·55-s + ⋯
L(s)  = 1  − 1.96·3-s + 0.275·5-s − 0.394·7-s + 2.87·9-s − 0.656·11-s − 0.758·13-s − 0.542·15-s + 0.242·17-s + 0.413·19-s + 0.777·21-s − 1.35·23-s − 0.923·25-s − 3.69·27-s + 0.769·29-s + 1.43·31-s + 1.29·33-s − 0.108·35-s + 1.99·37-s + 1.49·39-s + 1.15·41-s + 0.688·43-s + 0.792·45-s + 0.0654·47-s − 0.844·49-s − 0.477·51-s + 0.939·53-s − 0.180·55-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: \(1\)
Selberg data: \((2,\ 1088,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 10.2T + 27T^{2} \)
5 \( 1 - 3.08T + 125T^{2} \)
7 \( 1 + 7.31T + 343T^{2} \)
11 \( 1 + 23.9T + 1.33e3T^{2} \)
13 \( 1 + 35.5T + 2.19e3T^{2} \)
19 \( 1 - 34.2T + 6.85e3T^{2} \)
23 \( 1 + 149.T + 1.21e4T^{2} \)
29 \( 1 - 120.T + 2.43e4T^{2} \)
31 \( 1 - 247.T + 2.97e4T^{2} \)
37 \( 1 - 448.T + 5.06e4T^{2} \)
41 \( 1 - 303.T + 6.89e4T^{2} \)
43 \( 1 - 194.T + 7.95e4T^{2} \)
47 \( 1 - 21.0T + 1.03e5T^{2} \)
53 \( 1 - 362.T + 1.48e5T^{2} \)
59 \( 1 - 364.T + 2.05e5T^{2} \)
61 \( 1 + 478.T + 2.26e5T^{2} \)
67 \( 1 - 5.17T + 3.00e5T^{2} \)
71 \( 1 - 335.T + 3.57e5T^{2} \)
73 \( 1 + 1.08e3T + 3.89e5T^{2} \)
79 \( 1 + 561.T + 4.93e5T^{2} \)
83 \( 1 + 746.T + 5.71e5T^{2} \)
89 \( 1 - 1.11e3T + 7.04e5T^{2} \)
97 \( 1 - 247.T + 9.12e5T^{2} \)
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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

−9.636943915777853806593842058850, −7.925750763887971701159396890102, −7.27017091106400434376444247531, −6.12818818036513343144514201575, −5.91095626916583268644978375058, −4.85792758079194603747574098647, −4.16048911855976402078379384778, −2.45869618422995885067054643223, −1.01035587929278917691241898120, 0, 1.01035587929278917691241898120, 2.45869618422995885067054643223, 4.16048911855976402078379384778, 4.85792758079194603747574098647, 5.91095626916583268644978375058, 6.12818818036513343144514201575, 7.27017091106400434376444247531, 7.925750763887971701159396890102, 9.636943915777853806593842058850

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