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Properties

Label 1-2600-2600.667-r0-0-0
Degree $1$
Conductor $2600$
Sign $0.825 + 0.564i$
Analytic cond. $12.0743$
Root an. cond. $12.0743$
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  + (0.207 + 0.978i)3-s + (−0.866 − 0.5i)7-s + (−0.913 + 0.406i)9-s + (−0.913 − 0.406i)11-s + (0.207 − 0.978i)17-s + (−0.978 − 0.207i)19-s + (0.309 − 0.951i)21-s + (−0.406 + 0.913i)23-s + (−0.587 − 0.809i)27-s + (−0.978 + 0.207i)29-s + (0.309 + 0.951i)31-s + (0.207 − 0.978i)33-s + (−0.994 − 0.104i)37-s + (0.104 − 0.994i)41-s + (0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)3-s + (−0.866 − 0.5i)7-s + (−0.913 + 0.406i)9-s + (−0.913 − 0.406i)11-s + (0.207 − 0.978i)17-s + (−0.978 − 0.207i)19-s + (0.309 − 0.951i)21-s + (−0.406 + 0.913i)23-s + (−0.587 − 0.809i)27-s + (−0.978 + 0.207i)29-s + (0.309 + 0.951i)31-s + (0.207 − 0.978i)33-s + (−0.994 − 0.104i)37-s + (0.104 − 0.994i)41-s + (0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.825 + 0.564i$
Analytic conductor: \(12.0743\)
Root analytic conductor: \(12.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (0:\ ),\ 0.825 + 0.564i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9649288293 + 0.2980933129i\)
\(L(\frac12)\) \(\approx\) \(0.9649288293 + 0.2980933129i\)
\(L(1)\) \(\approx\) \(0.8327822059 + 0.1948853209i\)
\(L(1)\) \(\approx\) \(0.8327822059 + 0.1948853209i\)

Euler product

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

−19.096723693424287062014972335246, −18.73787629682004429615771498030, −18.087757519814241061815514239729, −17.1690219850494050682107007351, −16.663883684004439046713953220807, −15.59902734363733360805096765946, −15.04529144954098570252842412690, −14.32144515967227019384350523900, −13.3059126521288162846856557400, −12.86179988052782355105901669574, −12.40438459735335747209913951630, −11.58772984831812851537543587822, −10.55568928909115671918504864227, −9.956882580255827174585643728767, −8.91261634676915385636949662439, −8.35562963515238481196599557098, −7.59304603548521217547700947216, −6.79419545186938513932376045482, −6.03639549398097811399545163094, −5.55556270619095344399948686040, −4.25238428349219891944552809658, −3.37859515486717455983952900025, −2.34757724986202630037968330134, −2.033524511704114403957859133770, −0.55936512460273756448060841062, 0.53944573670179868734622016793, 2.166073854415195588226840095521, 3.01472399647661018245034240985, 3.63237313908254899518101505375, 4.44569813477275622136265966975, 5.3618328131868339820184669166, 5.92951510319967688723231878025, 7.07636360517629089682362674869, 7.71316154061700512359233282294, 8.79320153667195912527919707360, 9.23499945213662431655995721318, 10.221892808538757470991398495973, 10.52124386930948220737996944441, 11.34455138722130668667758313095, 12.27126367924714966142821454884, 13.18967372164670683493875528170, 13.76380922407435397215449667501, 14.423917689822484827839853773272, 15.466059425148008512314901849027, 15.83776056230214345946415722773, 16.44468697527277958372023105719, 17.157298124469175271156322277786, 17.9238765982817605526746990130, 18.98417308411025431411412895903, 19.43043763434043992484505310200

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