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Properties

Label 2-1088-17.4-c1-0-1
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: $\chi_{1088} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :1/2),\ -0.338 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6678646383\)
\(L(\frac12)\) \(\approx\) \(0.6678646383\)
\(L(\frac{3}{2})\) 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

−10.25610531028653299887484727588, −9.408826716068203497188671141106, −8.299909333331195667072495456309, −7.42308276127781704000541297178, −7.20336542672034962616550043028, −5.73867782289281987134174603357, −5.13959602663482749739626931886, −3.58765180102334901735722891931, −3.11600475752428243019535928750, −1.54441854597233720071674962369, 0.27802856210541519401078976973, 2.54485467259752839271024488022, 2.98992755669169397882472802008, 4.45842778508112805636220374855, 5.19882267970445842932066960791, 6.31940585623882120775439161018, 7.04697513736028133073443182273, 8.016587147637752169034165837811, 9.053438449255790387446388614599, 9.572569528480905445296975778503

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