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Properties

Label 2-1088-136.5-c0-0-1
Degree $2$
Conductor $1088$
Sign $0.581 + 0.813i$
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  + (1.63 − 1.08i)3-s + (1.08 − 2.63i)9-s + (−0.617 + 0.923i)11-s + (0.382 + 0.923i)17-s + (−1.30 + 0.541i)19-s + (−0.923 − 0.382i)25-s + (−0.707 − 3.55i)27-s + 2.17i·33-s + (0.324 + 1.63i)41-s + (−0.707 − 0.292i)43-s + (0.923 − 0.382i)49-s + (1.63 + 1.08i)51-s + (−1.54 + 2.30i)57-s + (0.292 − 0.707i)59-s − 1.84·67-s + ⋯
L(s)  = 1  + (1.63 − 1.08i)3-s + (1.08 − 2.63i)9-s + (−0.617 + 0.923i)11-s + (0.382 + 0.923i)17-s + (−1.30 + 0.541i)19-s + (−0.923 − 0.382i)25-s + (−0.707 − 3.55i)27-s + 2.17i·33-s + (0.324 + 1.63i)41-s + (−0.707 − 0.292i)43-s + (0.923 − 0.382i)49-s + (1.63 + 1.08i)51-s + (−1.54 + 2.30i)57-s + (0.292 − 0.707i)59-s − 1.84·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ 0.581 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.660445611\)
\(L(\frac12)\) \(\approx\) \(1.660445611\)
\(L(1)\) 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 \( 1 + (-1.63 + 1.08i)T + (0.382 - 0.923i)T^{2} \)
5 \( 1 + (0.923 + 0.382i)T^{2} \)
7 \( 1 + (-0.923 + 0.382i)T^{2} \)
11 \( 1 + (0.617 - 0.923i)T + (-0.382 - 0.923i)T^{2} \)
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 \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \)
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 \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \)
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 \( 1 + (-0.382 - 0.0761i)T + (0.923 + 0.382i)T^{2} \)
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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

−9.811613267907079841513398205360, −8.919005465229221409240164044162, −8.089614756027392973322370988120, −7.77169426451894519725598616307, −6.77999216196604105286836422323, −6.04150507285641789569719459309, −4.40154189429835938464567241474, −3.52044917408219854746246551470, −2.38532371052156721677025579578, −1.67704710944561599217617839097, 2.19561632444471797553493017367, 3.01665519258215574214678771693, 3.88378022165942806400041784304, 4.76522820644212633918526524619, 5.72063189645626981843233678532, 7.24106964416362259275122619329, 7.928388144139678080149535814949, 8.767201098452588763934713808527, 9.151128377921818821276928058117, 10.16277284767798291099300123164

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