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Properties

Label 1-5200-5200.2661-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.0929 - 0.995i$
Analytic cond. $24.1486$
Root an. cond. $24.1486$
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  + (0.994 + 0.104i)3-s + (0.5 − 0.866i)7-s + (0.978 + 0.207i)9-s + (−0.207 − 0.978i)11-s + (−0.104 − 0.994i)17-s + (−0.994 + 0.104i)19-s + (0.587 − 0.809i)21-s + (0.978 − 0.207i)23-s + (0.951 + 0.309i)27-s + (0.994 + 0.104i)29-s + (−0.809 + 0.587i)31-s + (−0.104 − 0.994i)33-s + (0.743 + 0.669i)37-s + (−0.669 + 0.743i)41-s + (−0.866 − 0.5i)43-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)3-s + (0.5 − 0.866i)7-s + (0.978 + 0.207i)9-s + (−0.207 − 0.978i)11-s + (−0.104 − 0.994i)17-s + (−0.994 + 0.104i)19-s + (0.587 − 0.809i)21-s + (0.978 − 0.207i)23-s + (0.951 + 0.309i)27-s + (0.994 + 0.104i)29-s + (−0.809 + 0.587i)31-s + (−0.104 − 0.994i)33-s + (0.743 + 0.669i)37-s + (−0.669 + 0.743i)41-s + (−0.866 − 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0929 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0929 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.0929 - 0.995i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (2661, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.0929 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.987044455 - 1.810207979i\)
\(L(\frac12)\) \(\approx\) \(1.987044455 - 1.810207979i\)
\(L(1)\) \(\approx\) \(1.516280136 - 0.3716657214i\)
\(L(1)\) \(\approx\) \(1.516280136 - 0.3716657214i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.994 + 0.104i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.207 - 0.978i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (-0.994 + 0.104i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (0.994 + 0.104i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.743 + 0.669i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.207 - 0.978i)T \)
61 \( 1 + (0.743 - 0.669i)T \)
67 \( 1 + (0.406 - 0.913i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−18.16350240609663138201988451218, −17.63651367682532342562186159285, −16.88801623196850347864030561360, −15.88829572696718794844292994459, −15.27684412247090705604317155280, −14.73322630577232792594781648135, −14.53802825432064274601704435984, −13.23710281101937878753429943518, −12.99725223868509886884895943656, −12.269641826779646830085141037059, −11.542842422070393803572407531861, −10.56622570437873735974644126783, −10.062152528181350128685904990083, −9.05657113703345995729023946587, −8.79288043082268773037269445080, −7.99416068209249952736324193028, −7.39602841382141762377093113152, −6.592744285561026670603535579, −5.804384037698289216963164338038, −4.79677287599834871010756472745, −4.30990610556040847017714761016, −3.39573982761136356704435179910, −2.468770485617470331953521548543, −2.046669116156794491443202961010, −1.24447314747508591502100744067, 0.58688769382673589693318890937, 1.47593689698231915702274527298, 2.356677389499420861514232169235, 3.22037081434308427590282216200, 3.678898714681146035452632109658, 4.76471459939741714719569086814, 5.018706812628911138073523468130, 6.42925572720769986476584299242, 6.95146500396698797359901084077, 7.70996145432358966560978335318, 8.52597181311035758883276947143, 8.69176328233550798870534300091, 9.838148919596560760738783919761, 10.290686174430057755091122857997, 11.10129507165203043194754439516, 11.62312875020797847314434287959, 12.83327983707954513219071391400, 13.25071208266728323994986410879, 13.89086794621915160294173169702, 14.4717803509696529527110279023, 14.987677345638418929036748803107, 15.85899473315440511870384808376, 16.46248316533412558294199989639, 17.020133375091399574233961386787, 17.96740439938700562365237572967

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