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Properties

Label 1-5600-5600.5059-r0-0-0
Degree $1$
Conductor $5600$
Sign $0.795 + 0.605i$
Analytic cond. $26.0062$
Root an. cond. $26.0062$
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  + (−0.629 + 0.777i)3-s + (−0.207 − 0.978i)9-s + (0.838 − 0.544i)11-s + (−0.891 + 0.453i)13-s + (−0.913 + 0.406i)17-s + (0.777 − 0.629i)19-s + (0.743 + 0.669i)23-s + (0.891 + 0.453i)27-s + (−0.987 − 0.156i)29-s + (0.913 − 0.406i)31-s + (−0.104 + 0.994i)33-s + (−0.544 + 0.838i)37-s + (0.207 − 0.978i)39-s + (−0.951 − 0.309i)41-s + (0.707 − 0.707i)43-s + ⋯
L(s)  = 1  + (−0.629 + 0.777i)3-s + (−0.207 − 0.978i)9-s + (0.838 − 0.544i)11-s + (−0.891 + 0.453i)13-s + (−0.913 + 0.406i)17-s + (0.777 − 0.629i)19-s + (0.743 + 0.669i)23-s + (0.891 + 0.453i)27-s + (−0.987 − 0.156i)29-s + (0.913 − 0.406i)31-s + (−0.104 + 0.994i)33-s + (−0.544 + 0.838i)37-s + (0.207 − 0.978i)39-s + (−0.951 − 0.309i)41-s + (0.707 − 0.707i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $0.795 + 0.605i$
Analytic conductor: \(26.0062\)
Root analytic conductor: \(26.0062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5600} (5059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5600,\ (0:\ ),\ 0.795 + 0.605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.159243977 + 0.3911170536i\)
\(L(\frac12)\) \(\approx\) \(1.159243977 + 0.3911170536i\)
\(L(1)\) \(\approx\) \(0.8421820363 + 0.1766793562i\)
\(L(1)\) \(\approx\) \(0.8421820363 + 0.1766793562i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.629 + 0.777i)T \)
11 \( 1 + (0.838 - 0.544i)T \)
13 \( 1 + (-0.891 + 0.453i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.777 - 0.629i)T \)
23 \( 1 + (0.743 + 0.669i)T \)
29 \( 1 + (-0.987 - 0.156i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.544 + 0.838i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (0.777 + 0.629i)T \)
59 \( 1 + (-0.998 - 0.0523i)T \)
61 \( 1 + (-0.998 + 0.0523i)T \)
67 \( 1 + (0.933 + 0.358i)T \)
71 \( 1 + (-0.587 - 0.809i)T \)
73 \( 1 + (0.207 - 0.978i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.156 - 0.987i)T \)
89 \( 1 + (0.743 + 0.669i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−17.77764802266092120155721189542, −17.10086933689689663045698935480, −16.75208844465436288658195840675, −15.858942008421065521003218919234, −15.126090596676678458525939573682, −14.37625268912210204977134723302, −13.8007551345890180991427031108, −13.00379119485384458898270293450, −12.38360980711103123786299717668, −11.94696462093637110815414120459, −11.2262149464170460224376851334, −10.54819574545041140267178821929, −9.77250635667942772292934495776, −9.055783773240706205435326836952, −8.2392737371783591292337062277, −7.36954569130770797814610880113, −7.011639358626658355425674203202, −6.31007250407391748354704265321, −5.44535074627133418140277211857, −4.880627724019182788234585805193, −4.10621748157141911500121800329, −3.00269239252540254227668744652, −2.24145102129846308126457412871, −1.47866962976006172611050016195, −0.57986613096619863722043240685, 0.60242860485310755913482155912, 1.590857891101260303522323365493, 2.68140626451531328820768717728, 3.494705960277006252931719067128, 4.19590472379849887874497022464, 4.86031469929826011497198743529, 5.52459666705126125897583851672, 6.334627344710989414721144215361, 6.90240044900225589281792707174, 7.68630406823787464155914554114, 8.92012224476572702578375325355, 9.131138417828372272418349565034, 9.84018015670869703996106234428, 10.71256778841948037445281617621, 11.20337014288075193506423340304, 11.92191679300862198817439661863, 12.27311759630850636367067771080, 13.537664750772862957238774848527, 13.80363971442149362196746885660, 14.99545925022929984845739782699, 15.16139738203913639247124270827, 15.949111184058215610600122444731, 16.72157543149068365101606172919, 17.28766284871759105333428113226, 17.48368537867258678780979722356

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