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Properties

Label 2-1096-1096.155-c0-0-0
Degree $2$
Conductor $1096$
Sign $0.658 - 0.752i$
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  + (−0.932 + 0.361i)2-s + (1.72 + 0.489i)3-s + (0.739 − 0.673i)4-s + (−1.78 + 0.165i)6-s + (−0.445 + 0.895i)8-s + (1.87 + 1.16i)9-s + (−0.136 + 0.124i)11-s + (1.60 − 0.798i)12-s + (0.0922 − 0.995i)16-s + (0.0822 + 0.165i)17-s + (−2.16 − 0.405i)18-s + (−0.876 − 0.163i)19-s + (0.0822 − 0.165i)22-s + (−1.20 + 1.32i)24-s + (−0.739 − 0.673i)25-s + ⋯
L(s)  = 1  + (−0.932 + 0.361i)2-s + (1.72 + 0.489i)3-s + (0.739 − 0.673i)4-s + (−1.78 + 0.165i)6-s + (−0.445 + 0.895i)8-s + (1.87 + 1.16i)9-s + (−0.136 + 0.124i)11-s + (1.60 − 0.798i)12-s + (0.0922 − 0.995i)16-s + (0.0822 + 0.165i)17-s + (−2.16 − 0.405i)18-s + (−0.876 − 0.163i)19-s + (0.0822 − 0.165i)22-s + (−1.20 + 1.32i)24-s + (−0.739 − 0.673i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.658 - 0.752i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ 0.658 - 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.201975918\)
\(L(\frac12)\) \(\approx\) \(1.201975918\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.932 - 0.361i)T \)
137 \( 1 + (-0.982 + 0.183i)T \)
good3 \( 1 + (-1.72 - 0.489i)T + (0.850 + 0.526i)T^{2} \)
5 \( 1 + (0.739 + 0.673i)T^{2} \)
7 \( 1 + (0.273 + 0.961i)T^{2} \)
11 \( 1 + (0.136 - 0.124i)T + (0.0922 - 0.995i)T^{2} \)
13 \( 1 + (-0.273 - 0.961i)T^{2} \)
17 \( 1 + (-0.0822 - 0.165i)T + (-0.602 + 0.798i)T^{2} \)
19 \( 1 + (0.876 + 0.163i)T + (0.932 + 0.361i)T^{2} \)
23 \( 1 + (-0.982 - 0.183i)T^{2} \)
29 \( 1 + (-0.982 - 0.183i)T^{2} \)
31 \( 1 + (0.932 - 0.361i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.92iT - T^{2} \)
43 \( 1 + (-0.365 + 1.95i)T + (-0.932 - 0.361i)T^{2} \)
47 \( 1 + (0.445 + 0.895i)T^{2} \)
53 \( 1 + (0.932 + 0.361i)T^{2} \)
59 \( 1 + (1.58 + 0.981i)T + (0.445 + 0.895i)T^{2} \)
61 \( 1 + (-0.445 + 0.895i)T^{2} \)
67 \( 1 + (-1.42 - 1.07i)T + (0.273 + 0.961i)T^{2} \)
71 \( 1 + (0.0922 + 0.995i)T^{2} \)
73 \( 1 + (0.329 + 0.436i)T + (-0.273 + 0.961i)T^{2} \)
79 \( 1 + (-0.850 + 0.526i)T^{2} \)
83 \( 1 + (1.42 + 0.711i)T + (0.602 + 0.798i)T^{2} \)
89 \( 1 + (0.132 - 0.342i)T + (-0.739 - 0.673i)T^{2} \)
97 \( 1 + (-0.907 - 0.995i)T + (-0.0922 + 0.995i)T^{2} \)
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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

−9.957802623755071108300280662123, −9.223889243504447756638785510876, −8.502225156697695278813509194326, −8.024893567750881222718006153950, −7.21881402193969029155201707372, −6.24425247175982353778466195463, −4.90083695951107430262401901631, −3.81943075293346178515708353464, −2.68098869046165183236369457980, −1.84938651237408403321539456709, 1.52001875603671119936875462236, 2.45641373273974074333005943518, 3.30448716955483206664452444508, 4.22028007057200902718862146825, 6.09258168822550246418780628506, 7.11408533617517451569357425695, 7.73276156339042079584423523374, 8.359931755053727079765498008601, 9.097582029810312248759761542977, 9.618075424340698041448534759460

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