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Properties

Label 2-1088-68.67-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.654421399\)
\(L(\frac12)\) \(\approx\) \(1.654421399\)
\(L(1)\) 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

−10.02898566247205208265839366686, −8.934126180989098613544342138941, −8.422079191721522123823316085560, −7.81023258946929560296350043444, −7.13906482954739713168556577335, −5.63923692441299072371902990987, −4.75548737432742474093321197366, −3.81767599940337080135518792609, −2.56039544138712227629001199825, −1.92173755837890980113294134007, 1.92173755837890980113294134007, 2.56039544138712227629001199825, 3.81767599940337080135518792609, 4.75548737432742474093321197366, 5.63923692441299072371902990987, 7.13906482954739713168556577335, 7.81023258946929560296350043444, 8.422079191721522123823316085560, 8.934126180989098613544342138941, 10.02898566247205208265839366686

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