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Properties

Label 2-1096-1096.563-c0-0-0
Degree $2$
Conductor $1096$
Sign $0.890 - 0.455i$
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  + (0.273 − 0.961i)2-s + (−0.247 + 1.32i)3-s + (−0.850 − 0.526i)4-s + (1.20 + 0.600i)6-s + (−0.739 + 0.673i)8-s + (−0.760 − 0.294i)9-s + (0.757 + 0.469i)11-s + (0.907 − 0.995i)12-s + (0.445 + 0.895i)16-s + (0.658 + 0.600i)17-s + (−0.491 + 0.650i)18-s + (−0.890 + 1.17i)19-s + (0.658 − 0.600i)22-s + (−0.709 − 1.14i)24-s + (0.850 − 0.526i)25-s + ⋯
L(s)  = 1  + (0.273 − 0.961i)2-s + (−0.247 + 1.32i)3-s + (−0.850 − 0.526i)4-s + (1.20 + 0.600i)6-s + (−0.739 + 0.673i)8-s + (−0.760 − 0.294i)9-s + (0.757 + 0.469i)11-s + (0.907 − 0.995i)12-s + (0.445 + 0.895i)16-s + (0.658 + 0.600i)17-s + (−0.491 + 0.650i)18-s + (−0.890 + 1.17i)19-s + (0.658 − 0.600i)22-s + (−0.709 − 1.14i)24-s + (0.850 − 0.526i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ 0.890 - 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9967075563\)
\(L(\frac12)\) \(\approx\) \(0.9967075563\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.273 + 0.961i)T \)
137 \( 1 + (-0.602 - 0.798i)T \)
good3 \( 1 + (0.247 - 1.32i)T + (-0.932 - 0.361i)T^{2} \)
5 \( 1 + (-0.850 + 0.526i)T^{2} \)
7 \( 1 + (0.982 - 0.183i)T^{2} \)
11 \( 1 + (-0.757 - 0.469i)T + (0.445 + 0.895i)T^{2} \)
13 \( 1 + (-0.982 + 0.183i)T^{2} \)
17 \( 1 + (-0.658 - 0.600i)T + (0.0922 + 0.995i)T^{2} \)
19 \( 1 + (0.890 - 1.17i)T + (-0.273 - 0.961i)T^{2} \)
23 \( 1 + (-0.602 + 0.798i)T^{2} \)
29 \( 1 + (-0.602 + 0.798i)T^{2} \)
31 \( 1 + (-0.273 + 0.961i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 0.367iT - T^{2} \)
43 \( 1 + (-1.42 - 1.07i)T + (0.273 + 0.961i)T^{2} \)
47 \( 1 + (0.739 + 0.673i)T^{2} \)
53 \( 1 + (-0.273 - 0.961i)T^{2} \)
59 \( 1 + (0.510 + 0.197i)T + (0.739 + 0.673i)T^{2} \)
61 \( 1 + (-0.739 + 0.673i)T^{2} \)
67 \( 1 + (-1.34 + 0.124i)T + (0.982 - 0.183i)T^{2} \)
71 \( 1 + (0.445 - 0.895i)T^{2} \)
73 \( 1 + (-0.181 + 1.95i)T + (-0.982 - 0.183i)T^{2} \)
79 \( 1 + (0.932 - 0.361i)T^{2} \)
83 \( 1 + (1.34 + 1.47i)T + (-0.0922 + 0.995i)T^{2} \)
89 \( 1 + (1.53 - 0.436i)T + (0.850 - 0.526i)T^{2} \)
97 \( 1 + (-0.554 + 0.895i)T + (-0.445 - 0.895i)T^{2} \)
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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

−10.23886328663164774064050948166, −9.592088383887552533240021642983, −8.897971378970042585634821499540, −7.957905433695130976321829518016, −6.39647816840504114997353743537, −5.58297876159824455566077655353, −4.54522477274029306383496691557, −4.06456044966126149707973542530, −3.15055236072186691438443432733, −1.66436693593590726062270934204, 0.971710179453613684431399728071, 2.69030679784203010628568649902, 3.97414505138500138707864702364, 5.12309383343602347680299447291, 6.00236679423872485018529270521, 6.81534924551980739653982650470, 7.17517201967322266203801172202, 8.161128091686016231741862136439, 8.858607937572830257793073026156, 9.685375891756913409876007116207

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