Properties

Label 2-1088-1.1-c3-0-2
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  + 1.23·3-s − 8.71·5-s − 33.7·7-s − 25.4·9-s − 57.0·11-s + 84.2·13-s − 10.7·15-s − 17·17-s − 159.·19-s − 41.7·21-s + 57.9·23-s − 49.1·25-s − 64.8·27-s − 124.·29-s + 98.3·31-s − 70.5·33-s + 294.·35-s − 355.·37-s + 104.·39-s − 67.6·41-s − 218.·43-s + 221.·45-s − 312.·47-s + 798.·49-s − 21.0·51-s + 148.·53-s + 497.·55-s + ⋯
L(s)  = 1  + 0.237·3-s − 0.779·5-s − 1.82·7-s − 0.943·9-s − 1.56·11-s + 1.79·13-s − 0.185·15-s − 0.242·17-s − 1.92·19-s − 0.433·21-s + 0.525·23-s − 0.392·25-s − 0.462·27-s − 0.798·29-s + 0.569·31-s − 0.371·33-s + 1.42·35-s − 1.57·37-s + 0.427·39-s − 0.257·41-s − 0.775·43-s + 0.735·45-s − 0.969·47-s + 2.32·49-s − 0.0576·51-s + 0.385·53-s + 1.21·55-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

\(L(2)\) \(\approx\) \(0.2922232454\)
\(L(\frac12)\) \(\approx\) \(0.2922232454\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.23T + 27T^{2} \)
5 \( 1 + 8.71T + 125T^{2} \)
7 \( 1 + 33.7T + 343T^{2} \)
11 \( 1 + 57.0T + 1.33e3T^{2} \)
13 \( 1 - 84.2T + 2.19e3T^{2} \)
19 \( 1 + 159.T + 6.85e3T^{2} \)
23 \( 1 - 57.9T + 1.21e4T^{2} \)
29 \( 1 + 124.T + 2.43e4T^{2} \)
31 \( 1 - 98.3T + 2.97e4T^{2} \)
37 \( 1 + 355.T + 5.06e4T^{2} \)
41 \( 1 + 67.6T + 6.89e4T^{2} \)
43 \( 1 + 218.T + 7.95e4T^{2} \)
47 \( 1 + 312.T + 1.03e5T^{2} \)
53 \( 1 - 148.T + 1.48e5T^{2} \)
59 \( 1 + 7.67T + 2.05e5T^{2} \)
61 \( 1 - 456.T + 2.26e5T^{2} \)
67 \( 1 - 47.9T + 3.00e5T^{2} \)
71 \( 1 - 786.T + 3.57e5T^{2} \)
73 \( 1 - 670.T + 3.89e5T^{2} \)
79 \( 1 + 34.1T + 4.93e5T^{2} \)
83 \( 1 - 322.T + 5.71e5T^{2} \)
89 \( 1 - 417.T + 7.04e5T^{2} \)
97 \( 1 - 996.T + 9.12e5T^{2} \)
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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

−9.382384107943896443247778121264, −8.477354124527153018961741779023, −8.159129136976948344378991443953, −6.82090210257370831107389638327, −6.23485434910940247588728455535, −5.32257852559203383238355337257, −3.82896648260912695663606824375, −3.34661774052204314689664592226, −2.33348000604658654115418910129, −0.25715121872298892775620576528, 0.25715121872298892775620576528, 2.33348000604658654115418910129, 3.34661774052204314689664592226, 3.82896648260912695663606824375, 5.32257852559203383238355337257, 6.23485434910940247588728455535, 6.82090210257370831107389638327, 8.159129136976948344378991443953, 8.477354124527153018961741779023, 9.382384107943896443247778121264

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