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Properties

Label 2-1088-136.29-c0-0-1
Degree $2$
Conductor $1088$
Sign $0.238 + 0.971i$
Analytic cond. $0.542982$
Root an. cond. $0.736873$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.216 − 0.324i)3-s + (0.324 − 0.783i)9-s + (−1.38 − 0.923i)11-s + (−0.382 − 0.923i)17-s + (1.30 − 0.541i)19-s + (0.923 + 0.382i)25-s + (−0.707 + 0.140i)27-s + 0.648i·33-s + (1.08 − 0.216i)41-s + (−0.707 − 0.292i)43-s + (−0.923 + 0.382i)49-s + (−0.216 + 0.324i)51-s + (−0.458 − 0.306i)57-s + (0.292 − 0.707i)59-s + 1.84·67-s + ⋯
L(s)  = 1  + (−0.216 − 0.324i)3-s + (0.324 − 0.783i)9-s + (−1.38 − 0.923i)11-s + (−0.382 − 0.923i)17-s + (1.30 − 0.541i)19-s + (0.923 + 0.382i)25-s + (−0.707 + 0.140i)27-s + 0.648i·33-s + (1.08 − 0.216i)41-s + (−0.707 − 0.292i)43-s + (−0.923 + 0.382i)49-s + (−0.216 + 0.324i)51-s + (−0.458 − 0.306i)57-s + (0.292 − 0.707i)59-s + 1.84·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $0.238 + 0.971i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ 0.238 + 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8797139916\)
\(L(\frac12)\) \(\approx\) \(0.8797139916\)
\(L(1)\) 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 \( 1 + (0.382 + 0.923i)T \)
good3 \( 1 + (0.216 + 0.324i)T + (-0.382 + 0.923i)T^{2} \)
5 \( 1 + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.382 + 0.923i)T^{2} \)
29 \( 1 + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \)
43 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.923 - 0.382i)T^{2} \)
67 \( 1 - 1.84T + T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
97 \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \)
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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

−9.786705000447735722274860916921, −9.164695370914321868254428512194, −8.173254256806412490504962180224, −7.33958442996363882097326745383, −6.63059854017392806911148272740, −5.55930958240600216466601131679, −4.90974906325448031316536393604, −3.48053993517281229400781267951, −2.63412474467905252652394098242, −0.858219503693463858791196716763, 1.82475968638737604559558867094, 2.97127828258223083986151844017, 4.32743604478598980164317574554, 5.03135021472081640512473953665, 5.82259110248044845426900940269, 7.06579342553269178922074673049, 7.75673316381749460058252560116, 8.474654626165852847224108540739, 9.723808542998295182430596368923, 10.20713327483401224405847813608

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