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Properties

Label 2-1096-1096.339-c0-0-0
Degree $2$
Conductor $1096$
Sign $0.724 + 0.688i$
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  + (−0.445 − 0.895i)2-s + (0.247 − 0.271i)3-s + (−0.602 + 0.798i)4-s + (−0.353 − 0.100i)6-s + (0.982 + 0.183i)8-s + (0.0798 + 0.861i)9-s + (−0.329 + 0.436i)11-s + (0.0675 + 0.361i)12-s + (−0.273 − 0.961i)16-s + (0.538 − 0.100i)17-s + (0.735 − 0.455i)18-s + (1.67 − 1.03i)19-s + (0.538 + 0.100i)22-s + (0.293 − 0.221i)24-s + (0.602 + 0.798i)25-s + ⋯
L(s)  = 1  + (−0.445 − 0.895i)2-s + (0.247 − 0.271i)3-s + (−0.602 + 0.798i)4-s + (−0.353 − 0.100i)6-s + (0.982 + 0.183i)8-s + (0.0798 + 0.861i)9-s + (−0.329 + 0.436i)11-s + (0.0675 + 0.361i)12-s + (−0.273 − 0.961i)16-s + (0.538 − 0.100i)17-s + (0.735 − 0.455i)18-s + (1.67 − 1.03i)19-s + (0.538 + 0.100i)22-s + (0.293 − 0.221i)24-s + (0.602 + 0.798i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.724 + 0.688i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ 0.724 + 0.688i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8860419874\)
\(L(\frac12)\) \(\approx\) \(0.8860419874\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.445 + 0.895i)T \)
137 \( 1 + (-0.850 - 0.526i)T \)
good3 \( 1 + (-0.247 + 0.271i)T + (-0.0922 - 0.995i)T^{2} \)
5 \( 1 + (-0.602 - 0.798i)T^{2} \)
7 \( 1 + (-0.739 + 0.673i)T^{2} \)
11 \( 1 + (0.329 - 0.436i)T + (-0.273 - 0.961i)T^{2} \)
13 \( 1 + (0.739 - 0.673i)T^{2} \)
17 \( 1 + (-0.538 + 0.100i)T + (0.932 - 0.361i)T^{2} \)
19 \( 1 + (-1.67 + 1.03i)T + (0.445 - 0.895i)T^{2} \)
23 \( 1 + (-0.850 + 0.526i)T^{2} \)
29 \( 1 + (-0.850 + 0.526i)T^{2} \)
31 \( 1 + (0.445 + 0.895i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.34iT - T^{2} \)
43 \( 1 + (1.01 + 1.63i)T + (-0.445 + 0.895i)T^{2} \)
47 \( 1 + (-0.982 + 0.183i)T^{2} \)
53 \( 1 + (0.445 - 0.895i)T^{2} \)
59 \( 1 + (-0.0822 - 0.887i)T + (-0.982 + 0.183i)T^{2} \)
61 \( 1 + (0.982 + 0.183i)T^{2} \)
67 \( 1 + (0.132 + 0.342i)T + (-0.739 + 0.673i)T^{2} \)
71 \( 1 + (-0.273 + 0.961i)T^{2} \)
73 \( 1 + (1.37 + 0.533i)T + (0.739 + 0.673i)T^{2} \)
79 \( 1 + (0.0922 - 0.995i)T^{2} \)
83 \( 1 + (-0.132 + 0.710i)T + (-0.932 - 0.361i)T^{2} \)
89 \( 1 + (0.942 + 0.469i)T + (0.602 + 0.798i)T^{2} \)
97 \( 1 + (-1.27 - 0.961i)T + (0.273 + 0.961i)T^{2} \)
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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

−10.02155616878217434548069768093, −9.228334243749989408480280685375, −8.443865674129092643658587948352, −7.50629478637366021205402223854, −7.13153433841358674541066662817, −5.35317039085094088334064793210, −4.74002011628767211317094477014, −3.37266090926151255070014700363, −2.55951836659513763532795767306, −1.36254890265494419215471625493, 1.17079301280459413130868254061, 3.09618917439939716827292703782, 4.10028031518199181296078926967, 5.26228658030191925192460329353, 5.97364743176308151978786581922, 6.87575132556651307765113659933, 7.78017405538561239767739615422, 8.437484654223565285604042455962, 9.318221876946087028974324260299, 9.908833579271248026158504295471

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