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Properties

Label 2-1088-68.67-c0-0-0
Degree $2$
Conductor $1088$
Sign $1$
Analytic cond. $0.542982$
Root an. cond. $0.736873$
Motivic weight $0$
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 − 1.41·7-s + 1.00·9-s + 1.41·11-s − 17-s + 2.00·21-s + 1.41·23-s + 25-s + 1.41·31-s − 2.00·33-s + 1.00·49-s + 1.41·51-s + 2·53-s − 1.41·63-s − 2.00·69-s − 1.41·71-s − 1.41·75-s − 2.00·77-s + 1.41·79-s − 0.999·81-s − 2.00·93-s + 1.41·99-s − 1.41·107-s + 1.41·119-s + ⋯
L(s)  = 1  − 1.41·3-s − 1.41·7-s + 1.00·9-s + 1.41·11-s − 17-s + 2.00·21-s + 1.41·23-s + 25-s + 1.41·31-s − 2.00·33-s + 1.00·49-s + 1.41·51-s + 2·53-s − 1.41·63-s − 2.00·69-s − 1.41·71-s − 1.41·75-s − 2.00·77-s + 1.41·79-s − 0.999·81-s − 2.00·93-s + 1.41·99-s − 1.41·107-s + 1.41·119-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: no
Arithmetic: yes
Character: $\chi_{1088} (1087, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5779876145\)
\(L(\frac12)\) \(\approx\) \(0.5779876145\)
\(L(1)\) 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 \( 1 + 1.41T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 2T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.21167418230782988748523401063, −9.296066038871673136654449650049, −8.713650738826188438592156819796, −6.99039660730034060091348377053, −6.66397939773053260846861252611, −6.04260430333767708789146036035, −4.97220783726061437243871088380, −4.05699066288755176613180912150, −2.87056178504683542145951033814, −0.941932617348772231688619127313, 0.941932617348772231688619127313, 2.87056178504683542145951033814, 4.05699066288755176613180912150, 4.97220783726061437243871088380, 6.04260430333767708789146036035, 6.66397939773053260846861252611, 6.99039660730034060091348377053, 8.713650738826188438592156819796, 9.296066038871673136654449650049, 10.21167418230782988748523401063

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