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Properties

Label 2-1096-1096.651-c0-0-0
Degree $2$
Conductor $1096$
Sign $0.181 - 0.983i$
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  + (−0.0922 + 0.995i)2-s + (−1.72 + 0.857i)3-s + (−0.982 − 0.183i)4-s + (−0.694 − 1.79i)6-s + (0.273 − 0.961i)8-s + (1.62 − 2.15i)9-s + (1.83 + 0.342i)11-s + (1.85 − 0.526i)12-s + (0.932 + 0.361i)16-s + (−0.510 − 1.79i)17-s + (1.99 + 1.81i)18-s + (−0.404 − 0.368i)19-s + (−0.510 + 1.79i)22-s + (0.353 + 1.89i)24-s + (0.982 − 0.183i)25-s + ⋯
L(s)  = 1  + (−0.0922 + 0.995i)2-s + (−1.72 + 0.857i)3-s + (−0.982 − 0.183i)4-s + (−0.694 − 1.79i)6-s + (0.273 − 0.961i)8-s + (1.62 − 2.15i)9-s + (1.83 + 0.342i)11-s + (1.85 − 0.526i)12-s + (0.932 + 0.361i)16-s + (−0.510 − 1.79i)17-s + (1.99 + 1.81i)18-s + (−0.404 − 0.368i)19-s + (−0.510 + 1.79i)22-s + (0.353 + 1.89i)24-s + (0.982 − 0.183i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.181 - 0.983i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ 0.181 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5513718878\)
\(L(\frac12)\) \(\approx\) \(0.5513718878\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0922 - 0.995i)T \)
137 \( 1 + (0.739 - 0.673i)T \)
good3 \( 1 + (1.72 - 0.857i)T + (0.602 - 0.798i)T^{2} \)
5 \( 1 + (-0.982 + 0.183i)T^{2} \)
7 \( 1 + (-0.445 + 0.895i)T^{2} \)
11 \( 1 + (-1.83 - 0.342i)T + (0.932 + 0.361i)T^{2} \)
13 \( 1 + (0.445 - 0.895i)T^{2} \)
17 \( 1 + (0.510 + 1.79i)T + (-0.850 + 0.526i)T^{2} \)
19 \( 1 + (0.404 + 0.368i)T + (0.0922 + 0.995i)T^{2} \)
23 \( 1 + (0.739 + 0.673i)T^{2} \)
29 \( 1 + (0.739 + 0.673i)T^{2} \)
31 \( 1 + (0.0922 - 0.995i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.79iT - T^{2} \)
43 \( 1 + (-0.486 + 0.533i)T + (-0.0922 - 0.995i)T^{2} \)
47 \( 1 + (-0.273 - 0.961i)T^{2} \)
53 \( 1 + (0.0922 + 0.995i)T^{2} \)
59 \( 1 + (0.111 - 0.147i)T + (-0.273 - 0.961i)T^{2} \)
61 \( 1 + (0.273 - 0.961i)T^{2} \)
67 \( 1 + (1.01 + 1.63i)T + (-0.445 + 0.895i)T^{2} \)
71 \( 1 + (0.932 - 0.361i)T^{2} \)
73 \( 1 + (-0.757 - 0.469i)T + (0.445 + 0.895i)T^{2} \)
79 \( 1 + (-0.602 - 0.798i)T^{2} \)
83 \( 1 + (-1.01 - 0.288i)T + (0.850 + 0.526i)T^{2} \)
89 \( 1 + (-1.34 + 0.124i)T + (0.982 - 0.183i)T^{2} \)
97 \( 1 + (-0.0675 + 0.361i)T + (-0.932 - 0.361i)T^{2} \)
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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

−10.10327809671077864387428222538, −9.325709492567668841588847407408, −8.982136174306419670885718090122, −7.30599635527956573602456864306, −6.60772457357184231207959190234, −6.22146166132555145610763309500, −4.97547924403056947035427464722, −4.65547510624977859396784893572, −3.69244047357773936716133462249, −0.894525661276553151792556640464, 1.10670958962211824246488351342, 1.93025855130957768895837953360, 3.83456726465005906344646849313, 4.55663126657184740860228204032, 5.76741158048506809501338670237, 6.30467171580797902411291036769, 7.20897685150131422840458227961, 8.358411137207465099242475572215, 9.158671339018408554770874493874, 10.37789602016048731448872850674

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