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Properties

Label 2-1088-17.16-c1-0-0
Degree $2$
Conductor $1088$
Sign $-0.928 - 0.371i$
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  + 2.61i·3-s − 3.69i·5-s + 1.08i·7-s − 3.82·9-s + 2.61i·11-s − 4.82·13-s + 9.65·15-s + (3.82 + 1.53i)17-s − 5.65·19-s − 2.82·21-s + 1.08i·23-s − 8.65·25-s − 2.16i·27-s + 6.75i·29-s + 8.47i·31-s + ⋯
L(s)  = 1  + 1.50i·3-s − 1.65i·5-s + 0.409i·7-s − 1.27·9-s + 0.787i·11-s − 1.33·13-s + 2.49·15-s + (0.928 + 0.371i)17-s − 1.29·19-s − 0.617·21-s + 0.225i·23-s − 1.73·25-s − 0.416i·27-s + 1.25i·29-s + 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $-0.928 - 0.371i$
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: \((2,\ 1088,\ (\ :1/2),\ -0.928 - 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8377575584\)
\(L(\frac12)\) \(\approx\) \(0.8377575584\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.82 - 1.53i)T \)
good3 \( 1 - 2.61iT - 3T^{2} \)
5 \( 1 + 3.69iT - 5T^{2} \)
7 \( 1 - 1.08iT - 7T^{2} \)
11 \( 1 - 2.61iT - 11T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 - 1.08iT - 23T^{2} \)
29 \( 1 - 6.75iT - 29T^{2} \)
31 \( 1 - 8.47iT - 31T^{2} \)
37 \( 1 - 3.69iT - 37T^{2} \)
41 \( 1 - 3.06iT - 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 + 9.65T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 1.65T + 59T^{2} \)
61 \( 1 + 8.02iT - 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + 10.6iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 + 3.24iT - 79T^{2} \)
83 \( 1 - 1.65T + 83T^{2} \)
89 \( 1 + 6.48T + 89T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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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

−9.875044863060568801576946210602, −9.630073697494201799664912474837, −8.671693978119644937693800329167, −8.222979146831928355720278177888, −6.86082241713096000053755395412, −5.40007683240586058007927224940, −4.97067426148081525908837484628, −4.42321362735413239902077571110, −3.29904150442042175538621166136, −1.73221396047931732195990122132, 0.35100307461567855603109937199, 2.15262971694462937867591620488, 2.75859687669961656276963686798, 3.96043893091164761081089663483, 5.60824664798972240055013725937, 6.45048382200009986371087225467, 6.97207574120224332420879429987, 7.67884781793366792245090997741, 8.228196658708977512878665146583, 9.719658220400046448237620061884

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