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Properties

Label 2-884-884.639-c0-0-0
Degree $2$
Conductor $884$
Sign $0.863 + 0.503i$
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 + (0.835 − 1.25i)5-s + (0.923 + 0.382i)8-s + (−0.991 − 0.130i)9-s + (0.483 + 1.42i)10-s + (0.608 − 0.793i)13-s + (−0.866 + 0.499i)16-s + (−0.965 − 0.258i)17-s + (0.707 − 0.707i)18-s + (−1.42 − 0.483i)20-s + (−0.482 − 1.16i)25-s + (0.258 + 0.965i)26-s + (1.42 − 1.24i)29-s + (0.130 − 0.991i)32-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)2-s + (−0.258 − 0.965i)4-s + (0.835 − 1.25i)5-s + (0.923 + 0.382i)8-s + (−0.991 − 0.130i)9-s + (0.483 + 1.42i)10-s + (0.608 − 0.793i)13-s + (−0.866 + 0.499i)16-s + (−0.965 − 0.258i)17-s + (0.707 − 0.707i)18-s + (−1.42 − 0.483i)20-s + (−0.482 − 1.16i)25-s + (0.258 + 0.965i)26-s + (1.42 − 1.24i)29-s + (0.130 − 0.991i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $0.863 + 0.503i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ 0.863 + 0.503i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7509710122\)
\(L(\frac12)\) \(\approx\) \(0.7509710122\)
\(L(1)\) 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 \( 1 + (-0.608 + 0.793i)T \)
17 \( 1 + (0.965 + 0.258i)T \)
good3 \( 1 + (0.991 + 0.130i)T^{2} \)
5 \( 1 + (-0.835 + 1.25i)T + (-0.382 - 0.923i)T^{2} \)
7 \( 1 + (-0.991 + 0.130i)T^{2} \)
11 \( 1 + (-0.793 + 0.608i)T^{2} \)
19 \( 1 + (-0.965 - 0.258i)T^{2} \)
23 \( 1 + (0.608 + 0.793i)T^{2} \)
29 \( 1 + (-1.42 + 1.24i)T + (0.130 - 0.991i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (0.0983 - 0.0862i)T + (0.130 - 0.991i)T^{2} \)
41 \( 1 + (0.793 + 0.391i)T + (0.608 + 0.793i)T^{2} \)
43 \( 1 + (0.965 + 0.258i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.758 - 1.83i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.258 + 0.965i)T^{2} \)
61 \( 1 + (-1.50 - 1.31i)T + (0.130 + 0.991i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.793 + 0.608i)T^{2} \)
73 \( 1 + (-1.49 - 0.996i)T + (0.382 + 0.923i)T^{2} \)
79 \( 1 + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.349 - 0.172i)T + (0.608 - 0.793i)T^{2} \)
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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

−10.00703210294493366025594841062, −9.179969298147605407654111182527, −8.581437952909565284204893213008, −8.088553249202946935109081915615, −6.72611399014290173089486543182, −5.86778218630162590600570312700, −5.33208828956986792277721616672, −4.33824375970648289767740601265, −2.45459317853292150650434710704, −0.955837364243948063021603711216, 1.88225025107850591427041501662, 2.75716269425626924879115403088, 3.66420416960154308070916116940, 5.06113449918347439749918003271, 6.45606846486162427633204089733, 6.82033299329452727943355017978, 8.197587367269571599343323054572, 8.843138514699933776900522301214, 9.690874069031635363260640139090, 10.53396424612564981933906400003

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