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Properties

Label 1-2600-2600.1531-r1-0-0
Degree $1$
Conductor $2600$
Sign $-0.456 - 0.889i$
Analytic cond. $279.408$
Root an. cond. $279.408$
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  + (0.669 + 0.743i)3-s + (−0.5 − 0.866i)7-s + (−0.104 + 0.994i)9-s + (0.104 + 0.994i)11-s + (0.669 − 0.743i)17-s + (−0.669 + 0.743i)19-s + (0.309 − 0.951i)21-s + (0.104 + 0.994i)23-s + (−0.809 + 0.587i)27-s + (−0.669 − 0.743i)29-s + (0.309 + 0.951i)31-s + (−0.669 + 0.743i)33-s + (0.913 − 0.406i)37-s + (−0.913 + 0.406i)41-s + (−0.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)3-s + (−0.5 − 0.866i)7-s + (−0.104 + 0.994i)9-s + (0.104 + 0.994i)11-s + (0.669 − 0.743i)17-s + (−0.669 + 0.743i)19-s + (0.309 − 0.951i)21-s + (0.104 + 0.994i)23-s + (−0.809 + 0.587i)27-s + (−0.669 − 0.743i)29-s + (0.309 + 0.951i)31-s + (−0.669 + 0.743i)33-s + (0.913 − 0.406i)37-s + (−0.913 + 0.406i)41-s + (−0.5 − 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $-0.456 - 0.889i$
Analytic conductor: \(279.408\)
Root analytic conductor: \(279.408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (1531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (1:\ ),\ -0.456 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05938035904 - 0.09715618616i\)
\(L(\frac12)\) \(\approx\) \(0.05938035904 - 0.09715618616i\)
\(L(1)\) \(\approx\) \(1.039985371 + 0.2893508169i\)
\(L(1)\) \(\approx\) \(1.039985371 + 0.2893508169i\)

Euler product

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

−19.35442487387448359600728875124, −18.63922435861304696936040930817, −18.44738650081885665029022536966, −17.29670217705675586415839965616, −16.64458022762370909202456350904, −15.8157494787244933209090153452, −14.853771676490523415174755426974, −14.66946155991352530529747781530, −13.53867557690364140546279532153, −13.037140564096148679940815229250, −12.422108840849058729466588754462, −11.66290259480966505662944502046, −10.86678900594900970485237443762, −9.82771321285907988108197065391, −9.068615091356851259926583513208, −8.47878595760494597646435534090, −7.92940410362125186245444904827, −6.83301719603406674293046183631, −6.22231005081344345048333677061, −5.637312544200460461474475308414, −4.390354708361185400446557981024, −3.36315602754527674408546427746, −2.81803579141939462768511554723, −1.98507046499027015040027427710, −0.96707320775091960895421161406, 0.01726669590804139313913420974, 1.3841434396598725838286565570, 2.29040094197892783955618019995, 3.329468618946957065777804550486, 3.85657515940472539468111814352, 4.65978392771796314968532676765, 5.42382919471964013053230315206, 6.538683278744340085433853705, 7.42048997799608332964945576573, 7.879893604448240440161565937478, 8.925749499930872201902635443000, 9.735822583262633004921123742440, 10.03501032405404079335344472639, 10.813892191310290078881084941884, 11.74650427090537631589923871323, 12.62100700640657008358040136999, 13.42429042546804823981353576022, 14.004183443797372362159729252384, 14.72391963680200303927529786785, 15.39895969953910351477550546164, 16.05576097443076980134318433017, 16.894646357027675431772518782149, 17.21874856062462290380678351864, 18.424595282216007736173895736106, 19.06874332606786933592673146656

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