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Properties

Label 1-2600-2600.947-r1-0-0
Degree $1$
Conductor $2600$
Sign $0.999 + 0.0242i$
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  + (−0.207 + 0.978i)3-s + (0.5 + 0.866i)7-s + (−0.913 − 0.406i)9-s + (−0.406 − 0.913i)11-s + (0.207 + 0.978i)17-s + (0.207 + 0.978i)19-s + (−0.951 + 0.309i)21-s + (−0.406 − 0.913i)23-s + (0.587 − 0.809i)27-s + (−0.978 − 0.207i)29-s + (−0.951 − 0.309i)31-s + (0.978 − 0.207i)33-s + (0.104 + 0.994i)37-s + (0.994 − 0.104i)41-s + (0.866 − 0.5i)43-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)3-s + (0.5 + 0.866i)7-s + (−0.913 − 0.406i)9-s + (−0.406 − 0.913i)11-s + (0.207 + 0.978i)17-s + (0.207 + 0.978i)19-s + (−0.951 + 0.309i)21-s + (−0.406 − 0.913i)23-s + (0.587 − 0.809i)27-s + (−0.978 − 0.207i)29-s + (−0.951 − 0.309i)31-s + (0.978 − 0.207i)33-s + (0.104 + 0.994i)37-s + (0.994 − 0.104i)41-s + (0.866 − 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.999 + 0.0242i$
Analytic conductor: \(279.408\)
Root analytic conductor: \(279.408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (947, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (1:\ ),\ 0.999 + 0.0242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.221714204 + 0.01479452223i\)
\(L(\frac12)\) \(\approx\) \(1.221714204 + 0.01479452223i\)
\(L(1)\) \(\approx\) \(0.8526114760 + 0.3044321056i\)
\(L(1)\) \(\approx\) \(0.8526114760 + 0.3044321056i\)

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 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.406 - 0.913i)T \)
17 \( 1 + (0.207 + 0.978i)T \)
19 \( 1 + (0.207 + 0.978i)T \)
23 \( 1 + (-0.406 - 0.913i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-0.951 - 0.309i)T \)
37 \( 1 + (0.104 + 0.994i)T \)
41 \( 1 + (0.994 - 0.104i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (-0.406 + 0.913i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (-0.669 - 0.743i)T \)
71 \( 1 + (-0.743 - 0.669i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.406 + 0.913i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−19.2779818336950358190441544213, −18.23740398729119136889930042027, −17.7986956919378472786266259676, −17.39362837657746216750921183901, −16.413676264515896573508336699182, −15.79495874772592741059947096073, −14.62881231088304485387537621720, −14.22260423777582449065636454879, −13.335397506823184600252339008678, −12.89384677430464706553548902275, −12.05959793618380476832063854219, −11.19783103062383508616653739919, −10.87274579072895864616092679946, −9.67887074977286941064409356487, −9.07661063333281603060095534850, −7.82945527883611919510371576494, −7.44423660712596151268173221753, −7.0106695861568301537242258091, −5.88461509021917495314970836891, −5.12790093966989309553424230775, −4.39782876007166414912266762608, −3.26647402311219859126596707306, −2.311891206937872494928040910007, −1.53498393262131745804961042570, −0.65446879384890983343228613843, 0.28471761064031484795938294916, 1.64665619791045726257694915530, 2.61459546333989161375047329468, 3.50978867313620421986523857481, 4.19404239913320958704758481554, 5.18882549027998786817873718194, 5.79301608537165695340402034236, 6.26380223245120831453346730095, 7.80047844874799118151412272602, 8.30785865665337196734678306429, 9.042896660860757001943009143104, 9.76022549520932250516245014845, 10.72987261357562327367409599312, 11.01525057622850443231149044924, 12.002340414886376020346845446565, 12.524764707700161733699440749638, 13.61226572329916075284797149569, 14.495313847240490399915731679551, 14.90627542169497842109982000331, 15.64425706757828564769318093226, 16.41552975150260713690896360513, 16.83895340233912788211857007507, 17.74901874889702188989571591351, 18.55898482802495477355016046709, 19.00655304789779754716534567494

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