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Properties

Label 1-2552-2552.565-r1-0-0
Degree $1$
Conductor $2552$
Sign $-0.609 - 0.792i$
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  + (−0.880 + 0.473i)3-s + (−0.995 − 0.0896i)5-s + (0.983 − 0.178i)7-s + (0.550 − 0.834i)9-s + (0.936 + 0.351i)13-s + (0.919 − 0.393i)15-s + (−0.951 − 0.309i)17-s + (−0.178 + 0.983i)19-s + (−0.781 + 0.623i)21-s + (−0.222 + 0.974i)23-s + (0.983 + 0.178i)25-s + (−0.0896 + 0.995i)27-s + (−0.512 − 0.858i)31-s + (−0.995 + 0.0896i)35-s + (−0.266 + 0.963i)37-s + ⋯
L(s)  = 1  + (−0.880 + 0.473i)3-s + (−0.995 − 0.0896i)5-s + (0.983 − 0.178i)7-s + (0.550 − 0.834i)9-s + (0.936 + 0.351i)13-s + (0.919 − 0.393i)15-s + (−0.951 − 0.309i)17-s + (−0.178 + 0.983i)19-s + (−0.781 + 0.623i)21-s + (−0.222 + 0.974i)23-s + (0.983 + 0.178i)25-s + (−0.0896 + 0.995i)27-s + (−0.512 − 0.858i)31-s + (−0.995 + 0.0896i)35-s + (−0.266 + 0.963i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2552 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2552 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2552\)    =    \(2^{3} \cdot 11 \cdot 29\)
Sign: $-0.609 - 0.792i$
Analytic conductor: \(274.250\)
Root analytic conductor: \(274.250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2552} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2552,\ (1:\ ),\ -0.609 - 0.792i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09208927813 - 0.1870637357i\)
\(L(\frac12)\) \(\approx\) \(0.09208927813 - 0.1870637357i\)
\(L(1)\) \(\approx\) \(0.6900737015 + 0.09232551863i\)
\(L(1)\) \(\approx\) \(0.6900737015 + 0.09232551863i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.880 + 0.473i)T \)
5 \( 1 + (-0.995 - 0.0896i)T \)
7 \( 1 + (0.983 - 0.178i)T \)
13 \( 1 + (0.936 + 0.351i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.178 + 0.983i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (-0.512 - 0.858i)T \)
37 \( 1 + (-0.266 + 0.963i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (-0.974 - 0.222i)T \)
47 \( 1 + (-0.266 - 0.963i)T \)
53 \( 1 + (-0.858 + 0.512i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.722 - 0.691i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (0.550 + 0.834i)T \)
73 \( 1 + (0.919 - 0.393i)T \)
79 \( 1 + (0.834 + 0.550i)T \)
83 \( 1 + (-0.134 - 0.990i)T \)
89 \( 1 + (-0.974 + 0.222i)T \)
97 \( 1 + (0.722 - 0.691i)T \)
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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

−19.53027026566123343858977071029, −18.54916797810206918232321756479, −18.03612621698613080292078167664, −17.60172918212158468668604894731, −16.5980962090260673968361126130, −15.969362775152582478665809517524, −15.342296124199358498355204101711, −14.594789691365465376633282813468, −13.66362536488349342715084002832, −12.83035980761769370666718841577, −12.295224286546529656127868405733, −11.37273604610046609738730647216, −11.008637166360593163344814859333, −10.57507167259602610739905611573, −9.099268494824919387298966160036, −8.3131924539675504576405143769, −7.836876983945043012184332617533, −6.86430140354631841338149980627, −6.353220082207013727537571004400, −5.26559356112767634953024740920, −4.634632518530429650294322275798, −3.95471066615583076953919294714, −2.72644345270795189147269636167, −1.72619072843828106968094133178, −0.82071179683450253595435599613, 0.05412669555710800797064439367, 1.07996396700737737767703285038, 1.94250195340246955146978347789, 3.57116964759642128962688185337, 3.96614107943786383647552196176, 4.769690625660467393196255408797, 5.45626342889521008786467216434, 6.404700347259177812394435532822, 7.17149456626524930270195528710, 8.029835967764343735236018932460, 8.67652179154299836796017206004, 9.59247058328256941928624011506, 10.55511525361916817349826416334, 11.23848300841011849571840351341, 11.52821773653965290135211115147, 12.25365477239921181325200932209, 13.177237431846603255385953001804, 14.05402258493479675083621010013, 14.954373325424187331885215182456, 15.5346680681335099418148277660, 16.07478058648038860953374869100, 16.89037728548311779386170581714, 17.41398622770616913861825005952, 18.43342389628346707407388698216, 18.62226367480504563215274255644

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