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Properties

Label 2-1096-1096.355-c0-0-0
Degree $2$
Conductor $1096$
Sign $-0.941 - 0.335i$
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  + (0.602 + 0.798i)2-s + (−0.719 − 0.0666i)3-s + (−0.273 + 0.961i)4-s + (−0.380 − 0.614i)6-s + (−0.932 + 0.361i)8-s + (−0.469 − 0.0878i)9-s + (−0.465 + 1.63i)11-s + (0.260 − 0.673i)12-s + (−0.850 − 0.526i)16-s + (−1.58 − 0.614i)17-s + (−0.213 − 0.427i)18-s + (0.831 + 1.66i)19-s + (−1.58 + 0.614i)22-s + (0.694 − 0.197i)24-s + (0.273 + 0.961i)25-s + ⋯
L(s)  = 1  + (0.602 + 0.798i)2-s + (−0.719 − 0.0666i)3-s + (−0.273 + 0.961i)4-s + (−0.380 − 0.614i)6-s + (−0.932 + 0.361i)8-s + (−0.469 − 0.0878i)9-s + (−0.465 + 1.63i)11-s + (0.260 − 0.673i)12-s + (−0.850 − 0.526i)16-s + (−1.58 − 0.614i)17-s + (−0.213 − 0.427i)18-s + (0.831 + 1.66i)19-s + (−1.58 + 0.614i)22-s + (0.694 − 0.197i)24-s + (0.273 + 0.961i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $-0.941 - 0.335i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ -0.941 - 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7539976454\)
\(L(\frac12)\) \(\approx\) \(0.7539976454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.602 - 0.798i)T \)
137 \( 1 + (0.445 - 0.895i)T \)
good3 \( 1 + (0.719 + 0.0666i)T + (0.982 + 0.183i)T^{2} \)
5 \( 1 + (-0.273 - 0.961i)T^{2} \)
7 \( 1 + (-0.0922 - 0.995i)T^{2} \)
11 \( 1 + (0.465 - 1.63i)T + (-0.850 - 0.526i)T^{2} \)
13 \( 1 + (0.0922 + 0.995i)T^{2} \)
17 \( 1 + (1.58 + 0.614i)T + (0.739 + 0.673i)T^{2} \)
19 \( 1 + (-0.831 - 1.66i)T + (-0.602 + 0.798i)T^{2} \)
23 \( 1 + (0.445 + 0.895i)T^{2} \)
29 \( 1 + (0.445 + 0.895i)T^{2} \)
31 \( 1 + (-0.602 - 0.798i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.99iT - T^{2} \)
43 \( 1 + (0.942 - 0.469i)T + (0.602 - 0.798i)T^{2} \)
47 \( 1 + (0.932 + 0.361i)T^{2} \)
53 \( 1 + (-0.602 + 0.798i)T^{2} \)
59 \( 1 + (-1.18 - 0.221i)T + (0.932 + 0.361i)T^{2} \)
61 \( 1 + (-0.932 + 0.361i)T^{2} \)
67 \( 1 + (-0.486 + 0.533i)T + (-0.0922 - 0.995i)T^{2} \)
71 \( 1 + (-0.850 + 0.526i)T^{2} \)
73 \( 1 + (0.136 - 0.124i)T + (0.0922 - 0.995i)T^{2} \)
79 \( 1 + (-0.982 + 0.183i)T^{2} \)
83 \( 1 + (0.486 + 1.25i)T + (-0.739 + 0.673i)T^{2} \)
89 \( 1 + (-1.42 - 1.07i)T + (0.273 + 0.961i)T^{2} \)
97 \( 1 + (-1.85 - 0.526i)T + (0.850 + 0.526i)T^{2} \)
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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

−10.47330337846931161094773793832, −9.482435520354991240098335308134, −8.682483949395693645756745341506, −7.59232434884791582234381051731, −7.04808610005220718907593465083, −6.16328362638302700060796312592, −5.28175850035191037108080531297, −4.70728098539524455893819769105, −3.56916352895887617172099935845, −2.23119689894166442956091697173, 0.59117339510943719327275878779, 2.45600446548811054995214865167, 3.28607484381595253514912090752, 4.58179724507289763187995628065, 5.25191409888270828712089512704, 6.13705657528426365975534045478, 6.73075997496556141524160811841, 8.423475415735040885660548286468, 8.841239707107876221226408127192, 10.02933416899605090589228910799

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