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Properties

Label 2-1088-17.4-c1-0-25
Degree $2$
Conductor $1088$
Sign $0.215 + 0.976i$
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  + (−1.57 + 1.57i)3-s + (2.77 − 2.77i)5-s + (−2.70 − 2.70i)7-s − 1.94i·9-s + (3.23 + 3.23i)11-s + 0.941·13-s + 8.73i·15-s + (−3.71 − 1.77i)17-s − 0.880i·19-s + 8.49·21-s + (−2.10 − 2.10i)23-s − 10.4i·25-s + (−1.66 − 1.66i)27-s + (1.71 − 1.71i)29-s + (4.36 − 4.36i)31-s + ⋯
L(s)  = 1  + (−0.907 + 0.907i)3-s + (1.24 − 1.24i)5-s + (−1.02 − 1.02i)7-s − 0.647i·9-s + (0.975 + 0.975i)11-s + 0.261·13-s + 2.25i·15-s + (−0.902 − 0.431i)17-s − 0.202i·19-s + 1.85·21-s + (−0.438 − 0.438i)23-s − 2.08i·25-s + (−0.320 − 0.320i)27-s + (0.319 − 0.319i)29-s + (0.784 − 0.784i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $0.215 + 0.976i$
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: \((2,\ 1088,\ (\ :1/2),\ 0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.064186933\)
\(L(\frac12)\) \(\approx\) \(1.064186933\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.71 + 1.77i)T \)
good3 \( 1 + (1.57 - 1.57i)T - 3iT^{2} \)
5 \( 1 + (-2.77 + 2.77i)T - 5iT^{2} \)
7 \( 1 + (2.70 + 2.70i)T + 7iT^{2} \)
11 \( 1 + (-3.23 - 3.23i)T + 11iT^{2} \)
13 \( 1 - 0.941T + 13T^{2} \)
19 \( 1 + 0.880iT - 19T^{2} \)
23 \( 1 + (2.10 + 2.10i)T + 23iT^{2} \)
29 \( 1 + (-1.71 + 1.71i)T - 29iT^{2} \)
31 \( 1 + (-4.36 + 4.36i)T - 31iT^{2} \)
37 \( 1 + (4.77 - 4.77i)T - 37iT^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 + 7.66iT - 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 7.55iT - 53T^{2} \)
59 \( 1 + 6.47iT - 59T^{2} \)
61 \( 1 + (0.778 + 0.778i)T + 61iT^{2} \)
67 \( 1 - 7.35T + 67T^{2} \)
71 \( 1 + (-11.2 + 11.2i)T - 71iT^{2} \)
73 \( 1 + (-10.4 + 10.4i)T - 73iT^{2} \)
79 \( 1 + (-1.22 - 1.22i)T + 79iT^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + (5.49 - 5.49i)T - 97iT^{2} \)
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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

−9.725532019020224990993139407835, −9.367476901724557690183065718787, −8.256316337321082775437122661924, −6.61718642961060363444397168544, −6.43925693184783011142549111944, −5.18585823318068819928510623934, −4.60396140309555776547090035087, −3.84325243559098072251204165457, −1.99389696548075917131948465808, −0.53674655081330202038428582091, 1.48347885996043195946689584761, 2.61075422254012229321654711046, 3.54648236161930162144161938805, 5.47490737346670346008906572119, 6.15863917548155767789173271343, 6.40930794150718464322371898592, 7.01807496652375920049062900615, 8.524155404611671827307393069587, 9.323812168984869172424760988999, 10.05829588810345780093760649428

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