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Properties

Label 1-2552-2552.1389-r1-0-0
Degree $1$
Conductor $2552$
Sign $0.735 - 0.677i$
Analytic cond. $274.250$
Root an. cond. $274.250$
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  + (−0.266 + 0.963i)3-s + (−0.691 − 0.722i)5-s + (−0.0448 − 0.998i)7-s + (−0.858 − 0.512i)9-s + (−0.995 − 0.0896i)13-s + (0.880 − 0.473i)15-s + (−0.951 + 0.309i)17-s + (0.998 + 0.0448i)19-s + (0.974 + 0.222i)21-s + (−0.900 − 0.433i)23-s + (−0.0448 + 0.998i)25-s + (0.722 − 0.691i)27-s + (−0.990 + 0.134i)31-s + (−0.691 + 0.722i)35-s + (0.657 + 0.753i)37-s + ⋯
L(s)  = 1  + (−0.266 + 0.963i)3-s + (−0.691 − 0.722i)5-s + (−0.0448 − 0.998i)7-s + (−0.858 − 0.512i)9-s + (−0.995 − 0.0896i)13-s + (0.880 − 0.473i)15-s + (−0.951 + 0.309i)17-s + (0.998 + 0.0448i)19-s + (0.974 + 0.222i)21-s + (−0.900 − 0.433i)23-s + (−0.0448 + 0.998i)25-s + (0.722 − 0.691i)27-s + (−0.990 + 0.134i)31-s + (−0.691 + 0.722i)35-s + (0.657 + 0.753i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2552 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2552 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2552\)    =    \(2^{3} \cdot 11 \cdot 29\)
Sign: $0.735 - 0.677i$
Analytic conductor: \(274.250\)
Root analytic conductor: \(274.250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2552} (1389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2552,\ (1:\ ),\ 0.735 - 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6289093072 - 0.2453006876i\)
\(L(\frac12)\) \(\approx\) \(0.6289093072 - 0.2453006876i\)
\(L(1)\) \(\approx\) \(0.6779470748 + 0.04578071948i\)
\(L(1)\) \(\approx\) \(0.6779470748 + 0.04578071948i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.266 + 0.963i)T \)
5 \( 1 + (-0.691 - 0.722i)T \)
7 \( 1 + (-0.0448 - 0.998i)T \)
13 \( 1 + (-0.995 - 0.0896i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (0.998 + 0.0448i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.990 + 0.134i)T \)
37 \( 1 + (0.657 + 0.753i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (0.657 - 0.753i)T \)
53 \( 1 + (-0.134 - 0.990i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.834 + 0.550i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (-0.858 + 0.512i)T \)
73 \( 1 + (0.880 - 0.473i)T \)
79 \( 1 + (-0.512 + 0.858i)T \)
83 \( 1 + (-0.936 - 0.351i)T \)
89 \( 1 + (-0.433 - 0.900i)T \)
97 \( 1 + (-0.834 + 0.550i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.22780982590071534326186036643, −18.58054600760156817034435745859, −18.0031416639635276654755365968, −17.51732570499412958205468128465, −16.39947634984913729373094464200, −15.74213496180223324603323827479, −15.03968199629194499490171745589, −14.25314175697978815329705481601, −13.69981213814110873424668119816, −12.50314649942798632229211660735, −12.27917893158570097994537895927, −11.3940869825988380907293299639, −11.04940013618692436346001880851, −9.832727323072483556297803430199, −9.04607020784779689551919474140, −8.17720418103318995546188612680, −7.3798254817628938516737061541, −7.02734618105537727170802802462, −5.99285490764641148558210363591, −5.43499051487871872727830128748, −4.39293880121296632666498534491, −3.26613258749609229206183721372, −2.45731573783841158023181012309, −1.93071986482923950198785630090, −0.450514048536674282522201288951, 0.246641805979407611172087092861, 1.17720562052493378636829086148, 2.57910514236637182856139691720, 3.62918826870430252566536587612, 4.21080022676956990270296421674, 4.80345857641367588150789153372, 5.545171152808659342546437570842, 6.63235888915809465463355877459, 7.483644211871299624610311746893, 8.21641488033410984357648261803, 9.03113895855131528975845978604, 9.8484786680671045883989921683, 10.31257653871885100847487091413, 11.38795618394498140738381226, 11.65258517868572232678896682896, 12.69961615777370601887569326216, 13.341732224911519859842640941980, 14.40556568661648230046630242587, 14.86650440886020360308645289730, 15.83706759862409724987077731577, 16.28183057038492747104701121658, 16.85827844869104019951298692067, 17.51927621516309637366526810424, 18.26795025048750822378475424310, 19.67570451529640856116105562140

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