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Properties

Label 2-1096-1096.323-c0-0-0
Degree $2$
Conductor $1096$
Sign $0.810 + 0.585i$
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  + (0.850 + 0.526i)2-s + (0.719 − 1.85i)3-s + (0.445 + 0.895i)4-s + (1.58 − 1.20i)6-s + (−0.0922 + 0.995i)8-s + (−2.19 − 1.99i)9-s + (0.537 + 1.07i)11-s + (1.98 − 0.183i)12-s + (−0.602 + 0.798i)16-s + (−0.111 − 1.20i)17-s + (−0.811 − 2.85i)18-s + (−0.0505 − 0.177i)19-s + (−0.111 + 1.20i)22-s + (1.78 + 0.887i)24-s + (−0.445 + 0.895i)25-s + ⋯
L(s)  = 1  + (0.850 + 0.526i)2-s + (0.719 − 1.85i)3-s + (0.445 + 0.895i)4-s + (1.58 − 1.20i)6-s + (−0.0922 + 0.995i)8-s + (−2.19 − 1.99i)9-s + (0.537 + 1.07i)11-s + (1.98 − 0.183i)12-s + (−0.602 + 0.798i)16-s + (−0.111 − 1.20i)17-s + (−0.811 − 2.85i)18-s + (−0.0505 − 0.177i)19-s + (−0.111 + 1.20i)22-s + (1.78 + 0.887i)24-s + (−0.445 + 0.895i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.810 + 0.585i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ 0.810 + 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.973012301\)
\(L(\frac12)\) \(\approx\) \(1.973012301\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.850 - 0.526i)T \)
137 \( 1 + (-0.273 + 0.961i)T \)
good3 \( 1 + (-0.719 + 1.85i)T + (-0.739 - 0.673i)T^{2} \)
5 \( 1 + (0.445 - 0.895i)T^{2} \)
7 \( 1 + (-0.932 + 0.361i)T^{2} \)
11 \( 1 + (-0.537 - 1.07i)T + (-0.602 + 0.798i)T^{2} \)
13 \( 1 + (0.932 - 0.361i)T^{2} \)
17 \( 1 + (0.111 + 1.20i)T + (-0.982 + 0.183i)T^{2} \)
19 \( 1 + (0.0505 + 0.177i)T + (-0.850 + 0.526i)T^{2} \)
23 \( 1 + (-0.273 - 0.961i)T^{2} \)
29 \( 1 + (-0.273 - 0.961i)T^{2} \)
31 \( 1 + (-0.850 - 0.526i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.722iT - T^{2} \)
43 \( 1 + (1.53 - 0.436i)T + (0.850 - 0.526i)T^{2} \)
47 \( 1 + (0.0922 + 0.995i)T^{2} \)
53 \( 1 + (-0.850 + 0.526i)T^{2} \)
59 \( 1 + (1.25 + 1.14i)T + (0.0922 + 0.995i)T^{2} \)
61 \( 1 + (-0.0922 + 0.995i)T^{2} \)
67 \( 1 + (-0.365 - 1.95i)T + (-0.932 + 0.361i)T^{2} \)
71 \( 1 + (-0.602 - 0.798i)T^{2} \)
73 \( 1 + (-1.83 - 0.342i)T + (0.932 + 0.361i)T^{2} \)
79 \( 1 + (0.739 - 0.673i)T^{2} \)
83 \( 1 + (0.365 + 0.0339i)T + (0.982 + 0.183i)T^{2} \)
89 \( 1 + (1.01 + 1.63i)T + (-0.445 + 0.895i)T^{2} \)
97 \( 1 + (-1.60 + 0.798i)T + (0.602 - 0.798i)T^{2} \)
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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

−9.627173868510510555639528929300, −8.823230810816695368386913569375, −7.981528837348467858954125127367, −7.22410115073266769450909923370, −6.87902865681954323286736141312, −6.01721891731909386927408477779, −4.93151817298229722266722241196, −3.57614824346578182096391027673, −2.62886308779763231105896341788, −1.68052297616302506048396751733, 2.19049591183431777267456024837, 3.35875382711051206301838025610, 3.83008076982472504528453979768, 4.65729409936571496746455348873, 5.59856342451980703264366079327, 6.30793529334242737062138098173, 8.010817121087572338494709644794, 8.774728205681963742800645320159, 9.480485152520525383116858469075, 10.37347013191142209608471709034

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