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Properties

Label 2-1088-17.13-c1-0-5
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: $\chi_{1088} (897, \cdot )$
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

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

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