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Properties

Label 2-1088-17.13-c1-0-27
Degree $2$
Conductor $1088$
Sign $-0.338 + 0.941i$
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  + (−0.431 − 0.431i)3-s + (−0.289 − 0.289i)5-s + (2.55 − 2.55i)7-s − 2.62i·9-s + (2.86 − 2.86i)11-s − 3.62·13-s + 0.249i·15-s + (3.91 − 1.28i)17-s + 6.83i·19-s − 2.20·21-s + (−5.84 + 5.84i)23-s − 4.83i·25-s + (−2.42 + 2.42i)27-s + (−5.91 − 5.91i)29-s + (−0.124 − 0.124i)31-s + ⋯
L(s)  = 1  + (−0.249 − 0.249i)3-s + (−0.129 − 0.129i)5-s + (0.965 − 0.965i)7-s − 0.875i·9-s + (0.862 − 0.862i)11-s − 1.00·13-s + 0.0644i·15-s + (0.949 − 0.312i)17-s + 1.56i·19-s − 0.481·21-s + (−1.21 + 1.21i)23-s − 0.966i·25-s + (−0.467 + 0.467i)27-s + (−1.09 − 1.09i)29-s + (−0.0224 − 0.0224i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $-0.338 + 0.941i$
Analytic conductor: \(8.68772\)
Root analytic conductor: \(2.94749\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :1/2),\ -0.338 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.405914834\)
\(L(\frac12)\) \(\approx\) \(1.405914834\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.91 + 1.28i)T \)
good3 \( 1 + (0.431 + 0.431i)T + 3iT^{2} \)
5 \( 1 + (0.289 + 0.289i)T + 5iT^{2} \)
7 \( 1 + (-2.55 + 2.55i)T - 7iT^{2} \)
11 \( 1 + (-2.86 + 2.86i)T - 11iT^{2} \)
13 \( 1 + 3.62T + 13T^{2} \)
19 \( 1 - 6.83iT - 19T^{2} \)
23 \( 1 + (5.84 - 5.84i)T - 23iT^{2} \)
29 \( 1 + (5.91 + 5.91i)T + 29iT^{2} \)
31 \( 1 + (0.124 + 0.124i)T + 31iT^{2} \)
37 \( 1 + (1.71 + 1.71i)T + 37iT^{2} \)
41 \( 1 + (-1 + i)T - 41iT^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 - 1.42iT - 53T^{2} \)
59 \( 1 - 5.72iT - 59T^{2} \)
61 \( 1 + (-2.28 + 2.28i)T - 61iT^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + (6.79 + 6.79i)T + 71iT^{2} \)
73 \( 1 + (4.83 + 4.83i)T + 73iT^{2} \)
79 \( 1 + (0.988 - 0.988i)T - 79iT^{2} \)
83 \( 1 + 7.44iT - 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 + (-5.20 - 5.20i)T + 97iT^{2} \)
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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

−9.753551398593614743661531888222, −8.747335966981179222254292052558, −7.70584077445190323382044840705, −7.39417156913488882119191779340, −6.09474431857775612111341315591, −5.51616107255341150840681928823, −4.08053532613940601470020641102, −3.66974823091055646972027061611, −1.81099980833574126242124022725, −0.66201312274845568139671427350, 1.74669665880527118746075731219, 2.64732465927441296521503604923, 4.23405182094333338637539820028, 4.97560564861339138144476957915, 5.61532167968171114554918383498, 6.90283139939848601168851430177, 7.63059987711093686705113678203, 8.478818596725945801590347724960, 9.345673612941965469723494969134, 10.07357090960375463886651761418

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