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Properties

Label 2-1096-1096.19-c0-0-0
Degree $2$
Conductor $1096$
Sign $-0.986 - 0.165i$
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  + (0.361 + 0.932i)2-s + (0.922 + 1.65i)3-s + (−0.739 + 0.673i)4-s + (−1.21 + 1.45i)6-s + (−0.895 − 0.445i)8-s + (−1.36 + 2.20i)9-s + (−0.124 − 0.136i)11-s + (−1.79 − 0.602i)12-s + (0.0922 − 0.995i)16-s + (1.78 − 0.887i)17-s + (−2.55 − 0.477i)18-s + (0.328 − 1.75i)19-s + (0.0822 − 0.165i)22-s + (−0.0875 − 1.89i)24-s + (−0.673 + 0.739i)25-s + ⋯
L(s)  = 1  + (0.361 + 0.932i)2-s + (0.922 + 1.65i)3-s + (−0.739 + 0.673i)4-s + (−1.21 + 1.45i)6-s + (−0.895 − 0.445i)8-s + (−1.36 + 2.20i)9-s + (−0.124 − 0.136i)11-s + (−1.79 − 0.602i)12-s + (0.0922 − 0.995i)16-s + (1.78 − 0.887i)17-s + (−2.55 − 0.477i)18-s + (0.328 − 1.75i)19-s + (0.0822 − 0.165i)22-s + (−0.0875 − 1.89i)24-s + (−0.673 + 0.739i)25-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $-0.986 - 0.165i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ -0.986 - 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478993401\)
\(L(\frac12)\) \(\approx\) \(1.478993401\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.361 - 0.932i)T \)
137 \( 1 + (0.183 + 0.982i)T \)
good3 \( 1 + (-0.922 - 1.65i)T + (-0.526 + 0.850i)T^{2} \)
5 \( 1 + (0.673 - 0.739i)T^{2} \)
7 \( 1 + (-0.273 - 0.961i)T^{2} \)
11 \( 1 + (0.124 + 0.136i)T + (-0.0922 + 0.995i)T^{2} \)
13 \( 1 + (0.961 - 0.273i)T^{2} \)
17 \( 1 + (-1.78 + 0.887i)T + (0.602 - 0.798i)T^{2} \)
19 \( 1 + (-0.328 + 1.75i)T + (-0.932 - 0.361i)T^{2} \)
23 \( 1 + (0.183 - 0.982i)T^{2} \)
29 \( 1 + (-0.183 + 0.982i)T^{2} \)
31 \( 1 + (0.361 + 0.932i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.688 - 0.688i)T + iT^{2} \)
43 \( 1 + (1.05 - 0.722i)T + (0.361 - 0.932i)T^{2} \)
47 \( 1 + (0.895 - 0.445i)T^{2} \)
53 \( 1 + (0.361 - 0.932i)T^{2} \)
59 \( 1 + (1.58 + 0.981i)T + (0.445 + 0.895i)T^{2} \)
61 \( 1 + (0.445 - 0.895i)T^{2} \)
67 \( 1 + (0.629 - 0.0878i)T + (0.961 - 0.273i)T^{2} \)
71 \( 1 + (-0.995 + 0.0922i)T^{2} \)
73 \( 1 + (1.15 + 1.53i)T + (-0.273 + 0.961i)T^{2} \)
79 \( 1 + (-0.526 - 0.850i)T^{2} \)
83 \( 1 + (0.0878 + 0.261i)T + (-0.798 + 0.602i)T^{2} \)
89 \( 1 + (-1.50 + 0.666i)T + (0.673 - 0.739i)T^{2} \)
97 \( 1 + (0.00426 - 0.0922i)T + (-0.995 - 0.0922i)T^{2} \)
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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

−10.05050619660785993520792805946, −9.389915475623553139858129013263, −8.994329185017940728883705973011, −7.906350443318989249892134186412, −7.45754745622702634883330973700, −5.97062950049176890692775945046, −5.04064998315836404830205825213, −4.59705443437471422905744624465, −3.36600538605782713844660177787, −2.93896933015366731934587076013, 1.27115318338210123242609476760, 2.05174781694976629357575093007, 3.20117121817872672418207404735, 3.86510584837712989305428879637, 5.64021203254660967811312549142, 6.12335757359798459595571277499, 7.40990970216863917642997577914, 8.078809291709858885792788516083, 8.689535126582935852766332551197, 9.790644833141634523295443130830

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