↑

Properties

Label 1-3009-3009.245-r0-0-0
Degree $1$
Conductor $3009$
Sign $0.124 - 0.992i$
Analytic cond. $13.9737$
Root an. cond. $13.9737$
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  + (0.583 − 0.812i)2-s + (−0.319 − 0.947i)4-s + (−0.201 + 0.979i)5-s + (−0.820 − 0.572i)7-s + (−0.955 − 0.293i)8-s + (0.677 + 0.735i)10-s + (−0.863 + 0.503i)11-s + (−0.515 + 0.856i)13-s + (−0.943 + 0.332i)14-s + (−0.796 + 0.605i)16-s + (0.626 + 0.779i)19-s + (0.992 − 0.121i)20-s + (−0.0946 + 0.995i)22-s + (−0.456 − 0.889i)23-s + (−0.918 − 0.395i)25-s + (0.395 + 0.918i)26-s + ⋯
L(s)  = 1  + (0.583 − 0.812i)2-s + (−0.319 − 0.947i)4-s + (−0.201 + 0.979i)5-s + (−0.820 − 0.572i)7-s + (−0.955 − 0.293i)8-s + (0.677 + 0.735i)10-s + (−0.863 + 0.503i)11-s + (−0.515 + 0.856i)13-s + (−0.943 + 0.332i)14-s + (−0.796 + 0.605i)16-s + (0.626 + 0.779i)19-s + (0.992 − 0.121i)20-s + (−0.0946 + 0.995i)22-s + (−0.456 − 0.889i)23-s + (−0.918 − 0.395i)25-s + (0.395 + 0.918i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3009\)    =    \(3 \cdot 17 \cdot 59\)
Sign: $0.124 - 0.992i$
Analytic conductor: \(13.9737\)
Root analytic conductor: \(13.9737\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3009} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3009,\ (0:\ ),\ 0.124 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8495716900 - 0.7499627740i\)
\(L(\frac12)\) \(\approx\) \(0.8495716900 - 0.7499627740i\)
\(L(1)\) \(\approx\) \(0.9166278586 - 0.3591905818i\)
\(L(1)\) \(\approx\) \(0.9166278586 - 0.3591905818i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.583 - 0.812i)T \)
5 \( 1 + (-0.201 + 0.979i)T \)
7 \( 1 + (-0.820 - 0.572i)T \)
11 \( 1 + (-0.863 + 0.503i)T \)
13 \( 1 + (-0.515 + 0.856i)T \)
19 \( 1 + (0.626 + 0.779i)T \)
23 \( 1 + (-0.456 - 0.889i)T \)
29 \( 1 + (-0.228 - 0.973i)T \)
31 \( 1 + (-0.480 - 0.877i)T \)
37 \( 1 + (-0.995 - 0.0946i)T \)
41 \( 1 + (0.306 + 0.951i)T \)
43 \( 1 + (-0.135 + 0.990i)T \)
47 \( 1 + (0.827 - 0.561i)T \)
53 \( 1 + (0.938 + 0.344i)T \)
61 \( 1 + (0.228 - 0.973i)T \)
67 \( 1 + (-0.468 - 0.883i)T \)
71 \( 1 + (0.549 - 0.835i)T \)
73 \( 1 + (0.332 + 0.943i)T \)
79 \( 1 + (-0.787 - 0.615i)T \)
83 \( 1 + (0.999 + 0.0270i)T \)
89 \( 1 + (0.986 + 0.161i)T \)
97 \( 1 + (0.432 + 0.901i)T \)
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

−19.28973343051599894340395164495, −18.30167897349515325964618925750, −17.685423013077757986157573531371, −16.97548011934340971242808164589, −16.1360841133625410560772358415, −15.748804961730526144009609484386, −15.36648985948190194620384074480, −14.294987549084178568860642095537, −13.40264281526327640251097624297, −13.08181029265440240505457437338, −12.27891008189453854839552762915, −11.912026736493578954372333309241, −10.71145284934647069652514574376, −9.747428523489533851980559553160, −8.87360492805923313155967212058, −8.562602767909944242133954261038, −7.49343006862399035275255516695, −7.09713351374376908294945517781, −5.77145384615454258151416602241, −5.459180581686319831169195557786, −4.90489827878735765082747183772, −3.69927198785349342402252964599, −3.16792960640092090830625372639, −2.235752941597158489772775670825, −0.63586618970174284559526733454, 0.43281987333838899848484269589, 1.931263300741688917411632306646, 2.500104260150701036680310385481, 3.35556299805820618430855011903, 4.01164081688512335817999436657, 4.73379115866779931376548709481, 5.83210750708540710536742686524, 6.44686187498831784574096120661, 7.23958331774523612636716693177, 7.95068832913137215572086270472, 9.31107318089707059138350841399, 9.91862738690582129122737120707, 10.35174280266280208822989657442, 11.092254626414267349037365490417, 11.89326261641911676458887102120, 12.468356791540606186965886505692, 13.32177907059299541369414941599, 13.893794945009367333617594342332, 14.55789693646744218524963610883, 15.20933297814916673099333654604, 15.962229651788189850396889409992, 16.72972057645287739011713554249, 17.764149965649005609252149271826, 18.62503469642570642671780957278, 18.80204986248863699972087081855

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