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Properties

Label 2-884-884.643-c0-0-0
Degree $2$
Conductor $884$
Sign $0.304 - 0.952i$
Analytic cond. $0.441173$
Root an. cond. $0.664208$
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  + (0.608 + 0.793i)2-s + (−0.258 + 0.965i)4-s + (0.172 − 0.867i)5-s + (−0.923 + 0.382i)8-s + (0.130 + 0.991i)9-s + (0.793 − 0.391i)10-s + (0.991 + 0.130i)13-s + (−0.866 − 0.499i)16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)18-s + (0.793 + 0.391i)20-s + (0.200 + 0.0832i)25-s + (0.499 + 0.866i)26-s + (−1.99 + 0.130i)29-s + (−0.130 − 0.991i)32-s + ⋯
L(s)  = 1  + (0.608 + 0.793i)2-s + (−0.258 + 0.965i)4-s + (0.172 − 0.867i)5-s + (−0.923 + 0.382i)8-s + (0.130 + 0.991i)9-s + (0.793 − 0.391i)10-s + (0.991 + 0.130i)13-s + (−0.866 − 0.499i)16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)18-s + (0.793 + 0.391i)20-s + (0.200 + 0.0832i)25-s + (0.499 + 0.866i)26-s + (−1.99 + 0.130i)29-s + (−0.130 − 0.991i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $0.304 - 0.952i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ 0.304 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.358425758\)
\(L(\frac12)\) \(\approx\) \(1.358425758\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.608 - 0.793i)T \)
13 \( 1 + (-0.991 - 0.130i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.130 - 0.991i)T^{2} \)
5 \( 1 + (-0.172 + 0.867i)T + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (-0.130 + 0.991i)T^{2} \)
11 \( 1 + (-0.608 + 0.793i)T^{2} \)
19 \( 1 + (-0.965 + 0.258i)T^{2} \)
23 \( 1 + (0.793 + 0.608i)T^{2} \)
29 \( 1 + (1.99 - 0.130i)T + (0.991 - 0.130i)T^{2} \)
31 \( 1 + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (0.123 + 1.88i)T + (-0.991 + 0.130i)T^{2} \)
41 \( 1 + (-0.423 + 1.24i)T + (-0.793 - 0.608i)T^{2} \)
43 \( 1 + (-0.965 + 0.258i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.46 - 0.607i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.258 - 0.965i)T^{2} \)
61 \( 1 + (0.641 + 0.0420i)T + (0.991 + 0.130i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.608 + 0.793i)T^{2} \)
73 \( 1 + (1.47 + 0.293i)T + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.357 - 1.05i)T + (-0.793 + 0.608i)T^{2} \)
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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

−10.63038467807385639793556650404, −9.309431591415866902458121939764, −8.711998585285193836734427299738, −7.86864725733781445425463369572, −7.18425609075866137454977793852, −5.84422331949970224443114575566, −5.45285403327947110884477993570, −4.38023400313882903340831300978, −3.52279275917946444087089262560, −1.87249548544910795489776618942, 1.39394239761761281693929727569, 2.96231531849742828239946534471, 3.49696703894536988239295008224, 4.66356128639516312017002677996, 5.90956200984528594504241725738, 6.40607435760418387892543371661, 7.46325635010494982023131103738, 8.785706130558894591691480161112, 9.633299193560252841916525116648, 10.20595730369329381016487104090

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