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Properties

Label 2-1096-1096.531-c0-0-0
Degree $2$
Conductor $1096$
Sign $0.959 - 0.280i$
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  + (−0.798 + 0.602i)2-s + (0.621 − 0.516i)3-s + (0.273 − 0.961i)4-s + (−0.184 + 0.786i)6-s + (0.361 + 0.932i)8-s + (−0.0638 + 0.341i)9-s + (1.63 + 0.465i)11-s + (−0.326 − 0.739i)12-s + (−0.850 − 0.526i)16-s + (0.380 − 0.981i)17-s + (−0.154 − 0.310i)18-s + (−0.646 + 0.322i)19-s + (−1.58 + 0.614i)22-s + (0.705 + 0.393i)24-s + (−0.961 + 0.273i)25-s + ⋯
L(s)  = 1  + (−0.798 + 0.602i)2-s + (0.621 − 0.516i)3-s + (0.273 − 0.961i)4-s + (−0.184 + 0.786i)6-s + (0.361 + 0.932i)8-s + (−0.0638 + 0.341i)9-s + (1.63 + 0.465i)11-s + (−0.326 − 0.739i)12-s + (−0.850 − 0.526i)16-s + (0.380 − 0.981i)17-s + (−0.154 − 0.310i)18-s + (−0.646 + 0.322i)19-s + (−1.58 + 0.614i)22-s + (0.705 + 0.393i)24-s + (−0.961 + 0.273i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.959 - 0.280i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ 0.959 - 0.280i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9348917970\)
\(L(\frac12)\) \(\approx\) \(0.9348917970\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.798 - 0.602i)T \)
137 \( 1 + (0.895 + 0.445i)T \)
good3 \( 1 + (-0.621 + 0.516i)T + (0.183 - 0.982i)T^{2} \)
5 \( 1 + (0.961 - 0.273i)T^{2} \)
7 \( 1 + (0.0922 + 0.995i)T^{2} \)
11 \( 1 + (-1.63 - 0.465i)T + (0.850 + 0.526i)T^{2} \)
13 \( 1 + (0.995 - 0.0922i)T^{2} \)
17 \( 1 + (-0.380 + 0.981i)T + (-0.739 - 0.673i)T^{2} \)
19 \( 1 + (0.646 - 0.322i)T + (0.602 - 0.798i)T^{2} \)
23 \( 1 + (0.895 - 0.445i)T^{2} \)
29 \( 1 + (-0.895 + 0.445i)T^{2} \)
31 \( 1 + (-0.798 + 0.602i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-1.08 + 1.08i)T - iT^{2} \)
43 \( 1 + (0.145 - 0.434i)T + (-0.798 - 0.602i)T^{2} \)
47 \( 1 + (-0.361 + 0.932i)T^{2} \)
53 \( 1 + (-0.798 - 0.602i)T^{2} \)
59 \( 1 + (-1.18 - 0.221i)T + (0.932 + 0.361i)T^{2} \)
61 \( 1 + (0.932 - 0.361i)T^{2} \)
67 \( 1 + (1.82 - 0.0844i)T + (0.995 - 0.0922i)T^{2} \)
71 \( 1 + (0.526 + 0.850i)T^{2} \)
73 \( 1 + (-1.47 + 1.34i)T + (0.0922 - 0.995i)T^{2} \)
79 \( 1 + (0.183 + 0.982i)T^{2} \)
83 \( 1 + (-0.0844 - 0.0373i)T + (0.673 + 0.739i)T^{2} \)
89 \( 1 + (0.629 - 0.0878i)T + (0.961 - 0.273i)T^{2} \)
97 \( 1 + (1.52 - 0.850i)T + (0.526 - 0.850i)T^{2} \)
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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

−9.740167411828933484900517313962, −9.176699973023612771250512467999, −8.438706821353090568591586292572, −7.59609648467387762490205899910, −7.01859399231082540595789981880, −6.18084135502896100658230379737, −5.13626226504128655628782739255, −3.93590952025300010479251955634, −2.40103075535493440104533366426, −1.43144418226973102478693315830, 1.34768981173461181066796454185, 2.71054035709796770440249827245, 3.81208667237123442698444291924, 4.16693711856930856151274453945, 6.06021571774354148196398632918, 6.73499015989547812602710627778, 7.966276690103979096371754495621, 8.597643330133405199163546809875, 9.280554681289013051643456671686, 9.793943546645124447488790111059

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