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Properties

Label 2-1096-1096.187-c0-0-0
Degree $2$
Conductor $1096$
Sign $0.848 + 0.529i$
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  + (0.739 − 0.673i)2-s + (1.02 + 0.634i)3-s + (0.0922 − 0.995i)4-s + (1.18 − 0.221i)6-s + (−0.602 − 0.798i)8-s + (0.201 + 0.405i)9-s + (−0.181 + 1.95i)11-s + (0.726 − 0.961i)12-s + (−0.982 − 0.183i)16-s + (1.18 − 1.56i)17-s + (0.422 + 0.163i)18-s + (−1.12 − 0.435i)19-s + (1.18 + 1.56i)22-s + (−0.111 − 1.20i)24-s + (0.0922 + 0.995i)25-s + ⋯
L(s)  = 1  + (0.739 − 0.673i)2-s + (1.02 + 0.634i)3-s + (0.0922 − 0.995i)4-s + (1.18 − 0.221i)6-s + (−0.602 − 0.798i)8-s + (0.201 + 0.405i)9-s + (−0.181 + 1.95i)11-s + (0.726 − 0.961i)12-s + (−0.982 − 0.183i)16-s + (1.18 − 1.56i)17-s + (0.422 + 0.163i)18-s + (−1.12 − 0.435i)19-s + (1.18 + 1.56i)22-s + (−0.111 − 1.20i)24-s + (0.0922 + 0.995i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.848 + 0.529i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ 0.848 + 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.980284307\)
\(L(\frac12)\) \(\approx\) \(1.980284307\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.739 + 0.673i)T \)
137 \( 1 + (-0.932 + 0.361i)T \)
good3 \( 1 + (-1.02 - 0.634i)T + (0.445 + 0.895i)T^{2} \)
5 \( 1 + (-0.0922 - 0.995i)T^{2} \)
7 \( 1 + (0.850 - 0.526i)T^{2} \)
11 \( 1 + (0.181 - 1.95i)T + (-0.982 - 0.183i)T^{2} \)
13 \( 1 + (0.850 - 0.526i)T^{2} \)
17 \( 1 + (-1.18 + 1.56i)T + (-0.273 - 0.961i)T^{2} \)
19 \( 1 + (1.12 + 0.435i)T + (0.739 + 0.673i)T^{2} \)
23 \( 1 + (-0.932 - 0.361i)T^{2} \)
29 \( 1 + (-0.932 - 0.361i)T^{2} \)
31 \( 1 + (-0.739 + 0.673i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.70T + T^{2} \)
43 \( 1 + (1.83 + 0.710i)T + (0.739 + 0.673i)T^{2} \)
47 \( 1 + (0.602 - 0.798i)T^{2} \)
53 \( 1 + (-0.739 - 0.673i)T^{2} \)
59 \( 1 + (-0.658 - 1.32i)T + (-0.602 + 0.798i)T^{2} \)
61 \( 1 + (0.602 + 0.798i)T^{2} \)
67 \( 1 + (-0.329 - 1.15i)T + (-0.850 + 0.526i)T^{2} \)
71 \( 1 + (0.982 - 0.183i)T^{2} \)
73 \( 1 + (-0.465 + 1.63i)T + (-0.850 - 0.526i)T^{2} \)
79 \( 1 + (-0.445 + 0.895i)T^{2} \)
83 \( 1 + (-0.329 - 0.436i)T + (-0.273 + 0.961i)T^{2} \)
89 \( 1 + (-1.37 - 1.25i)T + (0.0922 + 0.995i)T^{2} \)
97 \( 1 + (-0.0170 + 0.183i)T + (-0.982 - 0.183i)T^{2} \)
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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

−9.823145565359576709706652465858, −9.572972787423883389247447832709, −8.615388820370496632450451928522, −7.41294434952489909472545612008, −6.69281360012203874670204224682, −5.19331547954776126430887635581, −4.68415160213086787225276839868, −3.67284586237734270004861773553, −2.82693440072240029435746537960, −1.87049802787786218254120238909, 1.92685705512045231259223420904, 3.24355908106890123722911626895, 3.62903758184439242342475856511, 5.10212453182147010938219177405, 6.09829575646737996819934426636, 6.60946273344857190241930166229, 7.957045831658715201815805076234, 8.308129613714193991593408231577, 8.621162329337547814722370646481, 10.12074041150953214034156814995

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