Properties

Label 2-1096-1096.123-c0-0-0
Degree $2$
Conductor $1096$
Sign $-0.165 - 0.986i$
Analytic cond. $0.546975$
Root an. cond. $0.739577$
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  + (−0.850 + 0.526i)2-s + (0.172 − 0.0666i)3-s + (0.445 − 0.895i)4-s + (−0.111 + 0.147i)6-s + (0.0922 + 0.995i)8-s + (−0.713 + 0.650i)9-s + (−0.537 + 1.07i)11-s + (0.0170 − 0.183i)12-s + (−0.602 − 0.798i)16-s + (−0.111 + 1.20i)17-s + (0.264 − 0.929i)18-s + (−0.0505 + 0.177i)19-s + (−0.111 − 1.20i)22-s + (0.0822 + 0.165i)24-s + (0.445 + 0.895i)25-s + ⋯
L(s)  = 1  + (−0.850 + 0.526i)2-s + (0.172 − 0.0666i)3-s + (0.445 − 0.895i)4-s + (−0.111 + 0.147i)6-s + (0.0922 + 0.995i)8-s + (−0.713 + 0.650i)9-s + (−0.537 + 1.07i)11-s + (0.0170 − 0.183i)12-s + (−0.602 − 0.798i)16-s + (−0.111 + 1.20i)17-s + (0.264 − 0.929i)18-s + (−0.0505 + 0.177i)19-s + (−0.111 − 1.20i)22-s + (0.0822 + 0.165i)24-s + (0.445 + 0.895i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $-0.165 - 0.986i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ -0.165 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6064398663\)
\(L(\frac12)\) \(\approx\) \(0.6064398663\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.850 - 0.526i)T \)
137 \( 1 + (0.273 + 0.961i)T \)
good3 \( 1 + (-0.172 + 0.0666i)T + (0.739 - 0.673i)T^{2} \)
5 \( 1 + (-0.445 - 0.895i)T^{2} \)
7 \( 1 + (-0.932 - 0.361i)T^{2} \)
11 \( 1 + (0.537 - 1.07i)T + (-0.602 - 0.798i)T^{2} \)
13 \( 1 + (-0.932 - 0.361i)T^{2} \)
17 \( 1 + (0.111 - 1.20i)T + (-0.982 - 0.183i)T^{2} \)
19 \( 1 + (0.0505 - 0.177i)T + (-0.850 - 0.526i)T^{2} \)
23 \( 1 + (0.273 - 0.961i)T^{2} \)
29 \( 1 + (0.273 - 0.961i)T^{2} \)
31 \( 1 + (0.850 - 0.526i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.86T + T^{2} \)
43 \( 1 + (-0.329 + 1.15i)T + (-0.850 - 0.526i)T^{2} \)
47 \( 1 + (-0.0922 + 0.995i)T^{2} \)
53 \( 1 + (0.850 + 0.526i)T^{2} \)
59 \( 1 + (1.25 - 1.14i)T + (0.0922 - 0.995i)T^{2} \)
61 \( 1 + (-0.0922 - 0.995i)T^{2} \)
67 \( 1 + (0.181 + 0.0339i)T + (0.932 + 0.361i)T^{2} \)
71 \( 1 + (0.602 - 0.798i)T^{2} \)
73 \( 1 + (1.83 - 0.342i)T + (0.932 - 0.361i)T^{2} \)
79 \( 1 + (-0.739 - 0.673i)T^{2} \)
83 \( 1 + (0.181 + 1.95i)T + (-0.982 + 0.183i)T^{2} \)
89 \( 1 + (-0.465 - 0.288i)T + (0.445 + 0.895i)T^{2} \)
97 \( 1 + (-0.397 + 0.798i)T + (-0.602 - 0.798i)T^{2} \)
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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

−10.40058571061922409507977256280, −9.233808719718554175905033112795, −8.690677449773451892341322475953, −7.69892790881317944621534492892, −7.33130234431252070179051220306, −6.10475420900711391810546323365, −5.42413087129508980701811201466, −4.36527432661226470560705490679, −2.71539734692392795719339336614, −1.72318094269851665701458074779, 0.70969672337332284436241886612, 2.58631336643604946160904574929, 3.14023331360938676858256134926, 4.36908811002625152457734103735, 5.73268382029776558410482499904, 6.59856207938893759275552508043, 7.63144681429352397224344596816, 8.369751746290332415797257543970, 9.079145928851237313507963780715, 9.662348012140330476621246915423

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