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Properties

Label 2-1096-1096.667-c0-0-0
Degree $2$
Conductor $1096$
Sign $0.997 - 0.0684i$
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  + (0.932 − 0.361i)2-s + (−0.243 + 0.857i)3-s + (0.739 − 0.673i)4-s + (0.0822 + 0.887i)6-s + (0.445 − 0.895i)8-s + (0.174 + 0.108i)9-s + (0.136 − 0.124i)11-s + (0.397 + 0.798i)12-s + (0.0922 − 0.995i)16-s + (0.0822 + 0.165i)17-s + (0.201 + 0.0377i)18-s + (−0.876 − 0.163i)19-s + (0.0822 − 0.165i)22-s + (0.658 + 0.600i)24-s + (0.739 + 0.673i)25-s + ⋯
L(s)  = 1  + (0.932 − 0.361i)2-s + (−0.243 + 0.857i)3-s + (0.739 − 0.673i)4-s + (0.0822 + 0.887i)6-s + (0.445 − 0.895i)8-s + (0.174 + 0.108i)9-s + (0.136 − 0.124i)11-s + (0.397 + 0.798i)12-s + (0.0922 − 0.995i)16-s + (0.0822 + 0.165i)17-s + (0.201 + 0.0377i)18-s + (−0.876 − 0.163i)19-s + (0.0822 − 0.165i)22-s + (0.658 + 0.600i)24-s + (0.739 + 0.673i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.997 - 0.0684i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ 0.997 - 0.0684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.713659680\)
\(L(\frac12)\) \(\approx\) \(1.713659680\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.932 + 0.361i)T \)
137 \( 1 + (0.982 - 0.183i)T \)
good3 \( 1 + (0.243 - 0.857i)T + (-0.850 - 0.526i)T^{2} \)
5 \( 1 + (-0.739 - 0.673i)T^{2} \)
7 \( 1 + (0.273 + 0.961i)T^{2} \)
11 \( 1 + (-0.136 + 0.124i)T + (0.0922 - 0.995i)T^{2} \)
13 \( 1 + (0.273 + 0.961i)T^{2} \)
17 \( 1 + (-0.0822 - 0.165i)T + (-0.602 + 0.798i)T^{2} \)
19 \( 1 + (0.876 + 0.163i)T + (0.932 + 0.361i)T^{2} \)
23 \( 1 + (0.982 + 0.183i)T^{2} \)
29 \( 1 + (0.982 + 0.183i)T^{2} \)
31 \( 1 + (-0.932 + 0.361i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 0.547T + T^{2} \)
43 \( 1 + (0.181 + 0.0339i)T + (0.932 + 0.361i)T^{2} \)
47 \( 1 + (-0.445 - 0.895i)T^{2} \)
53 \( 1 + (-0.932 - 0.361i)T^{2} \)
59 \( 1 + (1.58 + 0.981i)T + (0.445 + 0.895i)T^{2} \)
61 \( 1 + (-0.445 + 0.895i)T^{2} \)
67 \( 1 + (0.537 - 0.711i)T + (-0.273 - 0.961i)T^{2} \)
71 \( 1 + (-0.0922 - 0.995i)T^{2} \)
73 \( 1 + (-0.329 - 0.436i)T + (-0.273 + 0.961i)T^{2} \)
79 \( 1 + (0.850 - 0.526i)T^{2} \)
83 \( 1 + (0.537 - 1.07i)T + (-0.602 - 0.798i)T^{2} \)
89 \( 1 + (1.83 + 0.710i)T + (0.739 + 0.673i)T^{2} \)
97 \( 1 + (-1.09 + 0.995i)T + (0.0922 - 0.995i)T^{2} \)
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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

−10.24492772374686881754730137690, −9.588298675199454208193331213424, −8.567707355987690391249597988322, −7.33088385165442078710903408629, −6.50553459357898108089437745561, −5.52052943205655912736913534244, −4.77164974289831215365165026919, −4.02366848191298718908676176370, −3.09620225533540777957652211369, −1.72777924855740039981321803901, 1.63239987118080670109143760219, 2.81305451611728840900098444520, 4.04206499599196719858455471337, 4.87632063814579581132506923720, 6.04979912912947248413412051731, 6.54597632777930636506985122107, 7.34881935926601282195574158400, 8.063245342180070598332474732930, 9.031286218878862707712114759852, 10.26491225884839166467941173967

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