Properties

Label 2-1088-1.1-c1-0-17
Degree $2$
Conductor $1088$
Sign $1$
Analytic cond. $8.68772$
Root an. cond. $2.94749$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s + 2·5-s + 4.24·7-s − 0.999·9-s + 1.41·11-s + 4·13-s + 2.82·15-s − 17-s − 2.82·19-s + 6·21-s − 4.24·23-s − 25-s − 5.65·27-s + 6·29-s − 7.07·31-s + 2.00·33-s + 8.48·35-s + 2·37-s + 5.65·39-s − 6·41-s + 8.48·43-s − 1.99·45-s − 11.3·47-s + 10.9·49-s − 1.41·51-s + 6·53-s + 2.82·55-s + ⋯
L(s)  = 1  + 0.816·3-s + 0.894·5-s + 1.60·7-s − 0.333·9-s + 0.426·11-s + 1.10·13-s + 0.730·15-s − 0.242·17-s − 0.648·19-s + 1.30·21-s − 0.884·23-s − 0.200·25-s − 1.08·27-s + 1.11·29-s − 1.27·31-s + 0.348·33-s + 1.43·35-s + 0.328·37-s + 0.905·39-s − 0.937·41-s + 1.29·43-s − 0.298·45-s − 1.65·47-s + 1.57·49-s − 0.198·51-s + 0.824·53-s + 0.381·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $1$
Analytic conductor: \(8.68772\)
Root analytic conductor: \(2.94749\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.870247839\)
\(L(\frac12)\) \(\approx\) \(2.870247839\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 7.07T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 - 7.07T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + 8T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−9.718408063080569721410448792298, −8.842235212805950074150496881617, −8.382799246385376257509583836301, −7.67360337367808583557597927630, −6.36925566900388697860019765014, −5.66332309008839671514364533785, −4.59996145185313663745225034559, −3.60772792012983346963502232962, −2.24512404465460750731967764302, −1.56211461648309840179014612238, 1.56211461648309840179014612238, 2.24512404465460750731967764302, 3.60772792012983346963502232962, 4.59996145185313663745225034559, 5.66332309008839671514364533785, 6.36925566900388697860019765014, 7.67360337367808583557597927630, 8.382799246385376257509583836301, 8.842235212805950074150496881617, 9.718408063080569721410448792298

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