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Properties

Label 2-1096-1096.11-c0-0-0
Degree $2$
Conductor $1096$
Sign $0.995 + 0.0952i$
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  + (0.995 − 0.0922i)2-s + (−1.65 + 0.555i)3-s + (0.982 − 0.183i)4-s + (−1.59 + 0.705i)6-s + (0.961 − 0.273i)8-s + (1.63 − 1.23i)9-s + (−0.342 − 1.83i)11-s + (−1.52 + 0.850i)12-s + (0.932 − 0.361i)16-s + (0.694 + 0.197i)17-s + (1.51 − 1.38i)18-s + (1.29 + 1.42i)19-s + (−0.510 − 1.79i)22-s + (−1.44 + 0.987i)24-s + (−0.183 + 0.982i)25-s + ⋯
L(s)  = 1  + (0.995 − 0.0922i)2-s + (−1.65 + 0.555i)3-s + (0.982 − 0.183i)4-s + (−1.59 + 0.705i)6-s + (0.961 − 0.273i)8-s + (1.63 − 1.23i)9-s + (−0.342 − 1.83i)11-s + (−1.52 + 0.850i)12-s + (0.932 − 0.361i)16-s + (0.694 + 0.197i)17-s + (1.51 − 1.38i)18-s + (1.29 + 1.42i)19-s + (−0.510 − 1.79i)22-s + (−1.44 + 0.987i)24-s + (−0.183 + 0.982i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.995 + 0.0952i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ 0.995 + 0.0952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.213972852\)
\(L(\frac12)\) \(\approx\) \(1.213972852\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.995 + 0.0922i)T \)
137 \( 1 + (-0.673 + 0.739i)T \)
good3 \( 1 + (1.65 - 0.555i)T + (0.798 - 0.602i)T^{2} \)
5 \( 1 + (0.183 - 0.982i)T^{2} \)
7 \( 1 + (0.445 + 0.895i)T^{2} \)
11 \( 1 + (0.342 + 1.83i)T + (-0.932 + 0.361i)T^{2} \)
13 \( 1 + (0.895 - 0.445i)T^{2} \)
17 \( 1 + (-0.694 - 0.197i)T + (0.850 + 0.526i)T^{2} \)
19 \( 1 + (-1.29 - 1.42i)T + (-0.0922 + 0.995i)T^{2} \)
23 \( 1 + (-0.673 - 0.739i)T^{2} \)
29 \( 1 + (0.673 + 0.739i)T^{2} \)
31 \( 1 + (0.995 - 0.0922i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-1.34 + 1.34i)T - iT^{2} \)
43 \( 1 + (1.82 + 0.0844i)T + (0.995 + 0.0922i)T^{2} \)
47 \( 1 + (-0.961 - 0.273i)T^{2} \)
53 \( 1 + (0.995 + 0.0922i)T^{2} \)
59 \( 1 + (0.111 + 0.147i)T + (-0.273 + 0.961i)T^{2} \)
61 \( 1 + (-0.273 - 0.961i)T^{2} \)
67 \( 1 + (-0.947 + 0.222i)T + (0.895 - 0.445i)T^{2} \)
71 \( 1 + (0.361 - 0.932i)T^{2} \)
73 \( 1 + (1.52 - 0.942i)T + (0.445 - 0.895i)T^{2} \)
79 \( 1 + (0.798 + 0.602i)T^{2} \)
83 \( 1 + (0.222 - 0.400i)T + (-0.526 - 0.850i)T^{2} \)
89 \( 1 + (1.53 - 1.27i)T + (0.183 - 0.982i)T^{2} \)
97 \( 1 + (1.36 + 0.932i)T + (0.361 + 0.932i)T^{2} \)
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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

−10.34912565061887763931277394756, −9.718298429388418920363203205255, −8.217338373748954325994828804237, −7.21002572858728137023809399415, −6.18760631765182994804191667571, −5.50589177456839139721559678998, −5.32473636306256324425957700839, −3.90231650476801727071623262143, −3.29395511043831776727659976449, −1.21873233778058410114164572376, 1.45747857775346533052359634254, 2.76679736746872555447931726669, 4.50835835576780447424959448235, 4.89427020389431781493790295576, 5.72861844682268251336072243171, 6.63483735897217488566805851759, 7.22266020672858288209188911667, 7.82019600909910076932198676804, 9.698053336313786415702494868851, 10.27124859015233126858828193430

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