↑

Properties

Label 1-2541-2541.2-r0-0-0
Degree $1$
Conductor $2541$
Sign $0.915 + 0.402i$
Analytic cond. $11.8003$
Root an. cond. $11.8003$
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  + (0.449 + 0.893i)2-s + (−0.595 + 0.803i)4-s + (0.532 − 0.846i)5-s + (−0.985 − 0.170i)8-s + (0.995 + 0.0950i)10-s + (0.870 − 0.491i)13-s + (−0.290 − 0.956i)16-s + (−0.179 + 0.983i)17-s + (−0.879 + 0.475i)19-s + (0.362 + 0.931i)20-s + (0.327 − 0.945i)23-s + (−0.432 − 0.901i)25-s + (0.830 + 0.556i)26-s + (−0.564 − 0.825i)29-s + (0.749 + 0.662i)31-s + (0.723 − 0.690i)32-s + ⋯
L(s)  = 1  + (0.449 + 0.893i)2-s + (−0.595 + 0.803i)4-s + (0.532 − 0.846i)5-s + (−0.985 − 0.170i)8-s + (0.995 + 0.0950i)10-s + (0.870 − 0.491i)13-s + (−0.290 − 0.956i)16-s + (−0.179 + 0.983i)17-s + (−0.879 + 0.475i)19-s + (0.362 + 0.931i)20-s + (0.327 − 0.945i)23-s + (−0.432 − 0.901i)25-s + (0.830 + 0.556i)26-s + (−0.564 − 0.825i)29-s + (0.749 + 0.662i)31-s + (0.723 − 0.690i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.915 + 0.402i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(11.8003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2541,\ (0:\ ),\ 0.915 + 0.402i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.042660368 + 0.4297740222i\)
\(L(\frac12)\) \(\approx\) \(2.042660368 + 0.4297740222i\)
\(L(1)\) \(\approx\) \(1.298798928 + 0.4147034171i\)
\(L(1)\) \(\approx\) \(1.298798928 + 0.4147034171i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.449 + 0.893i)T \)
5 \( 1 + (0.532 - 0.846i)T \)
13 \( 1 + (0.870 - 0.491i)T \)
17 \( 1 + (-0.179 + 0.983i)T \)
19 \( 1 + (-0.879 + 0.475i)T \)
23 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (-0.564 - 0.825i)T \)
31 \( 1 + (0.749 + 0.662i)T \)
37 \( 1 + (0.953 + 0.299i)T \)
41 \( 1 + (-0.254 - 0.967i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (0.935 + 0.353i)T \)
53 \( 1 + (0.290 - 0.956i)T \)
59 \( 1 + (0.710 + 0.703i)T \)
61 \( 1 + (-0.548 - 0.836i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (-0.610 + 0.791i)T \)
73 \( 1 + (-0.483 - 0.875i)T \)
79 \( 1 + (0.969 + 0.244i)T \)
83 \( 1 + (0.0855 - 0.996i)T \)
89 \( 1 + (-0.235 - 0.971i)T \)
97 \( 1 + (-0.466 + 0.884i)T \)
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

−19.358997182952377370785958203589, −18.6586094084113748247201616426, −18.24434896608605846955395178748, −17.50080143621316089788955741678, −16.61577486693713648275049095223, −15.46588734805201193522955306421, −15.03066285463054676609539136397, −14.13148850437364958369992723274, −13.56198406345705025396434230464, −13.13745473801793337918748089891, −12.06448889532967660955260972971, −11.25657445268832153109895199407, −10.94702551768354737108060580809, −10.0790654765628561327222693195, −9.318596256225763342128417796304, −8.803884318260429618423512063978, −7.511913636541138389777657500755, −6.62498154953749186462200657838, −5.98514230617161287932220423703, −5.171528704687337106392567656262, −4.23625652670254006570773092830, −3.457780577769840184766044532892, −2.63933495027331010203575898914, −1.97864036819764569571493172225, −0.95896414922107992693348689328, 0.68389729069312094629187088071, 1.87400194730923424502487861881, 2.95410827228504090585188165815, 4.08480090426619663019103244707, 4.47878646944467526397909310224, 5.57633997875409287239467150267, 6.03983312681341719451027031249, 6.69639543762824873065968550049, 7.89748582892744338637408355059, 8.44522871525070331689561954335, 8.92890728808972852393169275761, 9.938843779915922360923505458281, 10.7063906109556745033998434400, 11.81859065794650023305127820391, 12.68396150555193090166172792164, 13.02805228731563041486413476047, 13.69634324819400905761003688268, 14.56089819439651110316042871162, 15.18229601815147848001250407938, 15.97669085640469184790923514370, 16.58667573089213109692482597373, 17.246282734889374485538914410793, 17.7263493349072584061386508543, 18.59677219587615260449572792962, 19.37845452287572876128298795596

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