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Properties

Label 2-1096-1096.427-c0-0-0
Degree $2$
Conductor $1096$
Sign $-0.00417 - 0.999i$
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  + (−0.982 + 0.183i)2-s + (1.02 + 1.35i)3-s + (0.932 − 0.361i)4-s + (−1.25 − 1.14i)6-s + (−0.850 + 0.526i)8-s + (−0.517 + 1.81i)9-s + (1.37 − 0.533i)11-s + (1.44 + 0.895i)12-s + (0.739 − 0.673i)16-s + (−1.25 − 0.778i)17-s + (0.174 − 1.88i)18-s + (−0.156 + 1.69i)19-s + (−1.25 + 0.778i)22-s + (−1.58 − 0.614i)24-s + (0.932 + 0.361i)25-s + ⋯
L(s)  = 1  + (−0.982 + 0.183i)2-s + (1.02 + 1.35i)3-s + (0.932 − 0.361i)4-s + (−1.25 − 1.14i)6-s + (−0.850 + 0.526i)8-s + (−0.517 + 1.81i)9-s + (1.37 − 0.533i)11-s + (1.44 + 0.895i)12-s + (0.739 − 0.673i)16-s + (−1.25 − 0.778i)17-s + (0.174 − 1.88i)18-s + (−0.156 + 1.69i)19-s + (−1.25 + 0.778i)22-s + (−1.58 − 0.614i)24-s + (0.932 + 0.361i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00417 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00417 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $-0.00417 - 0.999i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ -0.00417 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9777266354\)
\(L(\frac12)\) \(\approx\) \(0.9777266354\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.982 - 0.183i)T \)
137 \( 1 + (-0.0922 - 0.995i)T \)
good3 \( 1 + (-1.02 - 1.35i)T + (-0.273 + 0.961i)T^{2} \)
5 \( 1 + (-0.932 - 0.361i)T^{2} \)
7 \( 1 + (0.602 - 0.798i)T^{2} \)
11 \( 1 + (-1.37 + 0.533i)T + (0.739 - 0.673i)T^{2} \)
13 \( 1 + (0.602 - 0.798i)T^{2} \)
17 \( 1 + (1.25 + 0.778i)T + (0.445 + 0.895i)T^{2} \)
19 \( 1 + (0.156 - 1.69i)T + (-0.982 - 0.183i)T^{2} \)
23 \( 1 + (-0.0922 + 0.995i)T^{2} \)
29 \( 1 + (-0.0922 + 0.995i)T^{2} \)
31 \( 1 + (0.982 - 0.183i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.20T + T^{2} \)
43 \( 1 + (-0.136 + 1.47i)T + (-0.982 - 0.183i)T^{2} \)
47 \( 1 + (0.850 + 0.526i)T^{2} \)
53 \( 1 + (0.982 + 0.183i)T^{2} \)
59 \( 1 + (-0.538 + 1.89i)T + (-0.850 - 0.526i)T^{2} \)
61 \( 1 + (0.850 - 0.526i)T^{2} \)
67 \( 1 + (0.757 + 1.52i)T + (-0.602 + 0.798i)T^{2} \)
71 \( 1 + (-0.739 - 0.673i)T^{2} \)
73 \( 1 + (0.537 - 1.07i)T + (-0.602 - 0.798i)T^{2} \)
79 \( 1 + (0.273 + 0.961i)T^{2} \)
83 \( 1 + (0.757 - 0.469i)T + (0.445 - 0.895i)T^{2} \)
89 \( 1 + (0.181 + 0.0339i)T + (0.932 + 0.361i)T^{2} \)
97 \( 1 + (-1.73 + 0.673i)T + (0.739 - 0.673i)T^{2} \)
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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

−9.968143199083202445878341515116, −9.366864321969927161111312537251, −8.733625722897102273439298458496, −8.285205941540885632385295393395, −7.11762461017689160286444787072, −6.21430896460404593436836316562, −5.02750812012471702979518053949, −3.89893080409618894982028704651, −3.11385580638220957285583734948, −1.83422373044318223314021507015, 1.24100160412843156705633585070, 2.17456120521688291589330692328, 3.05913684987743995240976645544, 4.34221671975304839275564068636, 6.35091565720047863621685258000, 6.78380746794905881199862535087, 7.34688523726527340654592184108, 8.485622504920083807178742210755, 8.804737961303592089096429430039, 9.454330614308975973095409976106

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