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Properties

Label 2-1088-17.16-c1-0-23
Degree $2$
Conductor $1088$
Sign $-0.242 + 0.970i$
Analytic cond. $8.68772$
Root an. cond. $2.94749$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·3-s + 2i·7-s − 9-s − 2i·11-s + 2·13-s + (1 − 4i)17-s − 4·19-s + 4·21-s − 6i·23-s + 5·25-s − 4i·27-s + 8i·29-s − 6i·31-s − 4·33-s − 8i·37-s + ⋯
L(s)  = 1  − 1.15i·3-s + 0.755i·7-s − 0.333·9-s − 0.603i·11-s + 0.554·13-s + (0.242 − 0.970i)17-s − 0.917·19-s + 0.872·21-s − 1.25i·23-s + 25-s − 0.769i·27-s + 1.48i·29-s − 1.07i·31-s − 0.696·33-s − 1.31i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $-0.242 + 0.970i$
Analytic conductor: \(8.68772\)
Root analytic conductor: \(2.94749\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :1/2),\ -0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.541267538\)
\(L(\frac12)\) \(\approx\) \(1.541267538\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 4i)T \)
good3 \( 1 + 2iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 + 6iT - 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.374418858124019989505542340837, −8.706177088265674913194129771735, −7.987506249200202309561689662332, −7.04936592979752097155340589229, −6.35776933667975455040783572202, −5.58715183230100239878567833612, −4.42100414246420826857384456341, −3.00628701979707347078973953694, −2.09205209022515955054758238205, −0.73578263655300909581935319684, 1.48747147094093378224399649572, 3.17202578170930542518847633559, 4.11408321295049290117647161137, 4.60964498021379795254527475322, 5.76632002832452695060741535666, 6.73218318843700346373603021408, 7.69032270334636273305722303884, 8.597650284848328467203909504584, 9.454751251620699662914830454252, 10.20498048192280451559818656002

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