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Properties

Label 2-1096-1096.259-c0-0-0
Degree $2$
Conductor $1096$
Sign $-0.984 - 0.175i$
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  + (−0.273 + 0.961i)2-s + (−1.45 − 0.271i)3-s + (−0.850 − 0.526i)4-s + (0.658 − 1.32i)6-s + (0.739 − 0.673i)8-s + (1.10 + 0.427i)9-s + (−0.757 − 0.469i)11-s + (1.09 + 0.995i)12-s + (0.445 + 0.895i)16-s + (0.658 + 0.600i)17-s + (−0.713 + 0.945i)18-s + (−0.890 + 1.17i)19-s + (0.658 − 0.600i)22-s + (−1.25 + 0.778i)24-s + (−0.850 + 0.526i)25-s + ⋯
L(s)  = 1  + (−0.273 + 0.961i)2-s + (−1.45 − 0.271i)3-s + (−0.850 − 0.526i)4-s + (0.658 − 1.32i)6-s + (0.739 − 0.673i)8-s + (1.10 + 0.427i)9-s + (−0.757 − 0.469i)11-s + (1.09 + 0.995i)12-s + (0.445 + 0.895i)16-s + (0.658 + 0.600i)17-s + (−0.713 + 0.945i)18-s + (−0.890 + 1.17i)19-s + (0.658 − 0.600i)22-s + (−1.25 + 0.778i)24-s + (−0.850 + 0.526i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $-0.984 - 0.175i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ -0.984 - 0.175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2167809961\)
\(L(\frac12)\) \(\approx\) \(0.2167809961\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.273 - 0.961i)T \)
137 \( 1 + (0.602 + 0.798i)T \)
good3 \( 1 + (1.45 + 0.271i)T + (0.932 + 0.361i)T^{2} \)
5 \( 1 + (0.850 - 0.526i)T^{2} \)
7 \( 1 + (0.982 - 0.183i)T^{2} \)
11 \( 1 + (0.757 + 0.469i)T + (0.445 + 0.895i)T^{2} \)
13 \( 1 + (0.982 - 0.183i)T^{2} \)
17 \( 1 + (-0.658 - 0.600i)T + (0.0922 + 0.995i)T^{2} \)
19 \( 1 + (0.890 - 1.17i)T + (-0.273 - 0.961i)T^{2} \)
23 \( 1 + (0.602 - 0.798i)T^{2} \)
29 \( 1 + (0.602 - 0.798i)T^{2} \)
31 \( 1 + (0.273 - 0.961i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.96T + T^{2} \)
43 \( 1 + (0.537 - 0.711i)T + (-0.273 - 0.961i)T^{2} \)
47 \( 1 + (-0.739 - 0.673i)T^{2} \)
53 \( 1 + (0.273 + 0.961i)T^{2} \)
59 \( 1 + (0.510 + 0.197i)T + (0.739 + 0.673i)T^{2} \)
61 \( 1 + (-0.739 + 0.673i)T^{2} \)
67 \( 1 + (-0.136 - 1.47i)T + (-0.982 + 0.183i)T^{2} \)
71 \( 1 + (-0.445 + 0.895i)T^{2} \)
73 \( 1 + (0.181 - 1.95i)T + (-0.982 - 0.183i)T^{2} \)
79 \( 1 + (-0.932 + 0.361i)T^{2} \)
83 \( 1 + (-0.136 + 0.124i)T + (0.0922 - 0.995i)T^{2} \)
89 \( 1 + (-0.329 - 1.15i)T + (-0.850 + 0.526i)T^{2} \)
97 \( 1 + (-1.44 - 0.895i)T + (0.445 + 0.895i)T^{2} \)
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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

−10.33722063626802852253287938475, −9.838523261771332381009589134793, −8.466146268030618926436641774338, −7.931960618329962285086279850746, −6.93219127116661810236917304947, −6.12853771392823011508073262440, −5.62836519729395657603793553993, −4.87034081807978195269338986413, −3.68078229839545080834855053639, −1.48490335673696699941274145361, 0.26862296337366910079794059438, 2.01779867421326781862094113416, 3.33821003955442541377206475295, 4.71205253721195397570327476201, 4.98587622776501197705299678649, 6.13258195257156181022027226954, 7.16918265078425905067530007067, 8.134270394190440285881388333622, 9.144301411091850496674285651101, 10.13517934579757684398847306189

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