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Properties

Label 2-1092-273.62-c1-0-24
Degree $2$
Conductor $1092$
Sign $0.831 + 0.555i$
Analytic cond. $8.71966$
Root an. cond. $2.95290$
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  + (1.5 − 0.866i)3-s + (−2 + 1.73i)7-s + (1.5 − 2.59i)9-s + (3.5 + 0.866i)13-s + (4 − 6.92i)19-s + (−1.50 + 4.33i)21-s + 5·25-s − 5.19i·27-s + 11·31-s + (−6 + 3.46i)37-s + (6 − 1.73i)39-s + (2.5 − 4.33i)43-s + (1.00 − 6.92i)49-s − 13.8i·57-s + (13.5 + 7.79i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.755 + 0.654i)7-s + (0.5 − 0.866i)9-s + (0.970 + 0.240i)13-s + (0.917 − 1.58i)19-s + (−0.327 + 0.944i)21-s + 25-s − 0.999i·27-s + 1.97·31-s + (−0.986 + 0.569i)37-s + (0.960 − 0.277i)39-s + (0.381 − 0.660i)43-s + (0.142 − 0.989i)49-s − 1.83i·57-s + (1.72 + 0.997i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1092\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.831 + 0.555i$
Analytic conductor: \(8.71966\)
Root analytic conductor: \(2.95290\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1092} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1092,\ (\ :1/2),\ 0.831 + 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.181521683\)
\(L(\frac12)\) \(\approx\) \(2.181521683\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (2 - 1.73i)T \)
13 \( 1 + (-3.5 - 0.866i)T \)
good5 \( 1 - 5T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 11T + 31T^{2} \)
37 \( 1 + (6 - 3.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-13.5 - 7.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.5 - 7.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + 17T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.5 - 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.586199541711910202214014872255, −8.790478430932363851871607071033, −8.448865485031426985165341189169, −7.12039458000625421847563285571, −6.67782336195005654106809721920, −5.65971459000901902155134722260, −4.41167949470973011374539746557, −3.19532949058457497195563845719, −2.60955596672108982244408104147, −1.07587453092469376932737508591, 1.31652987430929555875672861968, 2.93804530096129584342711799468, 3.59771323118935979648626018799, 4.47729519958010979001313675890, 5.68444071835514067100070661748, 6.66267969184645634505730482586, 7.62367980088880712619583310525, 8.337574288240794133232047733644, 9.127447279968069129830724161666, 10.11011635862176435243696133360

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