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Properties

Label 2-1088-68.67-c0-0-1
Degree $2$
Conductor $1088$
Sign $1$
Analytic cond. $0.542982$
Root an. cond. $0.736873$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s + 2·13-s + 17-s + 25-s − 49-s − 2·53-s + 81-s − 2·89-s + 2·101-s − 2·117-s + ⋯
L(s)  = 1  − 9-s + 2·13-s + 17-s + 25-s − 49-s − 2·53-s + 81-s − 2·89-s + 2·101-s − 2·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $1$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1088} (1087, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066456845\)
\(L(\frac12)\) \(\approx\) \(1.066456845\)
\(L(1)\) 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 \( 1 - T \)
good3 \( 1 + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( ( 1 - T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 + T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 + T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−10.11607116313888964352218061541, −9.081802150278103874746541284819, −8.473192560861006315063926554132, −7.77307745955764230151618591072, −6.50780021151692892930741056642, −5.92395760898668046421472765951, −4.99491168444129834428163193342, −3.69379665731245283156746754868, −2.96339081761551840908085493334, −1.33674598123762693440695948513, 1.33674598123762693440695948513, 2.96339081761551840908085493334, 3.69379665731245283156746754868, 4.99491168444129834428163193342, 5.92395760898668046421472765951, 6.50780021151692892930741056642, 7.77307745955764230151618591072, 8.473192560861006315063926554132, 9.081802150278103874746541284819, 10.11607116313888964352218061541

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