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Properties

Label 2-1096-1096.1035-c0-0-0
Degree $2$
Conductor $1096$
Sign $-0.671 + 0.741i$
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  + (0.895 − 0.445i)2-s + (−0.0762 − 1.64i)3-s + (0.602 − 0.798i)4-s + (−0.802 − 1.44i)6-s + (0.183 − 0.982i)8-s + (−1.71 + 0.158i)9-s + (0.436 + 0.329i)11-s + (−1.36 − 0.932i)12-s + (−0.273 − 0.961i)16-s + (0.353 + 1.89i)17-s + (−1.46 + 0.906i)18-s + (−0.193 − 0.312i)19-s + (0.538 + 0.100i)22-s + (−1.63 − 0.227i)24-s + (−0.798 + 0.602i)25-s + ⋯
L(s)  = 1  + (0.895 − 0.445i)2-s + (−0.0762 − 1.64i)3-s + (0.602 − 0.798i)4-s + (−0.802 − 1.44i)6-s + (0.183 − 0.982i)8-s + (−1.71 + 0.158i)9-s + (0.436 + 0.329i)11-s + (−1.36 − 0.932i)12-s + (−0.273 − 0.961i)16-s + (0.353 + 1.89i)17-s + (−1.46 + 0.906i)18-s + (−0.193 − 0.312i)19-s + (0.538 + 0.100i)22-s + (−1.63 − 0.227i)24-s + (−0.798 + 0.602i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $-0.671 + 0.741i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (1035, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ -0.671 + 0.741i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.672069390\)
\(L(\frac12)\) \(\approx\) \(1.672069390\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.895 + 0.445i)T \)
137 \( 1 + (0.526 - 0.850i)T \)
good3 \( 1 + (0.0762 + 1.64i)T + (-0.995 + 0.0922i)T^{2} \)
5 \( 1 + (0.798 - 0.602i)T^{2} \)
7 \( 1 + (0.739 - 0.673i)T^{2} \)
11 \( 1 + (-0.436 - 0.329i)T + (0.273 + 0.961i)T^{2} \)
13 \( 1 + (-0.673 - 0.739i)T^{2} \)
17 \( 1 + (-0.353 - 1.89i)T + (-0.932 + 0.361i)T^{2} \)
19 \( 1 + (0.193 + 0.312i)T + (-0.445 + 0.895i)T^{2} \)
23 \( 1 + (0.526 + 0.850i)T^{2} \)
29 \( 1 + (-0.526 - 0.850i)T^{2} \)
31 \( 1 + (0.895 - 0.445i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.0653 + 0.0653i)T - iT^{2} \)
43 \( 1 + (-0.947 - 0.222i)T + (0.895 + 0.445i)T^{2} \)
47 \( 1 + (-0.183 - 0.982i)T^{2} \)
53 \( 1 + (0.895 + 0.445i)T^{2} \)
59 \( 1 + (-0.0822 - 0.887i)T + (-0.982 + 0.183i)T^{2} \)
61 \( 1 + (-0.982 - 0.183i)T^{2} \)
67 \( 1 + (-0.456 + 1.03i)T + (-0.673 - 0.739i)T^{2} \)
71 \( 1 + (0.961 + 0.273i)T^{2} \)
73 \( 1 + (1.25 + 0.486i)T + (0.739 + 0.673i)T^{2} \)
79 \( 1 + (-0.995 - 0.0922i)T^{2} \)
83 \( 1 + (1.03 + 1.50i)T + (-0.361 + 0.932i)T^{2} \)
89 \( 1 + (-1.84 + 0.618i)T + (0.798 - 0.602i)T^{2} \)
97 \( 1 + (1.96 - 0.273i)T + (0.961 - 0.273i)T^{2} \)
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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

−9.959160721792330967956190655351, −8.822981685636177617882115936241, −7.78951742671120109483325171125, −7.13980921360850462351948086356, −6.16751697609955436654468946014, −5.85361693448084130570173882016, −4.45326344558681417188676791623, −3.36214445706690402944853982217, −2.10837884132794237412423973040, −1.37374613614094738649182727481, 2.63307841881709686484974621059, 3.57363361204990447541206983515, 4.31244237762518642375417495852, 5.12094482496372819406520166525, 5.78436019047997864525931983130, 6.82552480072628932078078969316, 7.87773094332664439728436528953, 8.818032437665067068107455928049, 9.590386649665023068954017606567, 10.32964316363954989662976376221

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