↑

Properties

Label 1-5200-5200.1539-r0-0-0
Degree $1$
Conductor $5200$
Sign $-0.991 + 0.129i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 − 0.809i)3-s − i·7-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.809 + 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (0.309 − 0.951i)23-s + (−0.951 − 0.309i)27-s + (0.587 − 0.809i)29-s + (−0.587 − 0.809i)31-s + (0.587 + 0.809i)33-s + (−0.309 − 0.951i)37-s + (−0.951 + 0.309i)41-s + i·43-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)3-s − i·7-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.809 + 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (0.309 − 0.951i)23-s + (−0.951 − 0.309i)27-s + (0.587 − 0.809i)29-s + (−0.587 − 0.809i)31-s + (0.587 + 0.809i)33-s + (−0.309 − 0.951i)37-s + (−0.951 + 0.309i)41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $-0.991 + 0.129i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (1539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ -0.991 + 0.129i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07992335686 - 1.226781889i\)
\(L(\frac12)\) \(\approx\) \(-0.07992335686 - 1.226781889i\)
\(L(1)\) \(\approx\) \(0.9667031273 - 0.5404709456i\)
\(L(1)\) \(\approx\) \(0.9667031273 - 0.5404709456i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.587 - 0.809i)T \)
31 \( 1 + (-0.587 - 0.809i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.951 + 0.309i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.587 + 0.809i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.951 - 0.309i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.43387848460566450693494518491, −17.74643813687780521971385769699, −16.76909787826071541405419607961, −16.18164080369939870488920115837, −15.5830561416754504946769344394, −15.24951606548552976197183523119, −14.293690762836926825218673982995, −13.78664781014416879273265883295, −13.22968126360025037036306746181, −12.15218976411441034322151017813, −11.65268411430717091833587803616, −10.8115763410853397101176864277, −10.311845622308197181857754178, −9.29871334141486896516556204898, −8.98638418274138353834623309120, −8.35996493039643682312201648493, −7.60455750647036723036420630228, −6.72001775087877028629903981900, −5.636441854638799189765450575752, −5.30789972065845407321682438764, −4.54066025341579351331094628677, −3.3880469345879908521709857527, −3.136229745310391297293769782966, −2.26986126620116591640200430959, −1.34524733628835261292337959736, 0.28647931282161635898034456402, 1.201767119923469531474521062601, 2.10722661546617059952200740835, 2.67039546632050515701339952940, 3.712812718302723050969926345739, 4.287321375423720999937931013534, 5.1406167969562738956297169195, 6.19468911934199377932084513925, 6.959934663762710419340373653649, 7.237400374567260830343620445851, 8.09416108500989546916496853756, 8.65177316173868617678505679839, 9.59798411763153850159660148192, 10.07928050875068947937733790331, 11.002563422532921183151875314980, 11.62489044642649168099645480760, 12.54649046740663448235146825907, 13.02976417332573026724568820147, 13.56072262828459568632334460442, 14.2341076063263817484434552022, 14.927172773592623890558187545964, 15.46059891563758867719285642823, 16.372875914839976925773609072712, 17.17778752495263507679685368010, 17.71907443390175801605070620769

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