Properties

Label 1-6001-6001.111-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.241 + 0.970i$
Analytic cond. $27.8685$
Root an. cond. $27.8685$
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  + (0.909 + 0.415i)2-s + (−0.909 + 0.415i)3-s + (0.654 + 0.755i)4-s + (−0.909 + 0.415i)5-s − 6-s i·7-s + (0.281 + 0.959i)8-s + (0.654 − 0.755i)9-s − 10-s + (−0.212 + 0.977i)11-s + (−0.909 − 0.415i)12-s + (0.800 − 0.599i)13-s + (0.415 − 0.909i)14-s + (0.654 − 0.755i)15-s + (−0.142 + 0.989i)16-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)2-s + (−0.909 + 0.415i)3-s + (0.654 + 0.755i)4-s + (−0.909 + 0.415i)5-s − 6-s i·7-s + (0.281 + 0.959i)8-s + (0.654 − 0.755i)9-s − 10-s + (−0.212 + 0.977i)11-s + (−0.909 − 0.415i)12-s + (0.800 − 0.599i)13-s + (0.415 − 0.909i)14-s + (0.654 − 0.755i)15-s + (−0.142 + 0.989i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.241 + 0.970i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.241 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.475297485 + 1.153675012i\)
\(L(\frac12)\) \(\approx\) \(1.475297485 + 1.153675012i\)
\(L(1)\) \(\approx\) \(1.100680931 + 0.5027423148i\)
\(L(1)\) \(\approx\) \(1.100680931 + 0.5027423148i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (0.909 + 0.415i)T \)
3 \( 1 + (-0.909 + 0.415i)T \)
5 \( 1 + (-0.909 + 0.415i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.212 + 0.977i)T \)
13 \( 1 + (0.800 - 0.599i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (-0.212 - 0.977i)T \)
29 \( 1 + (-0.997 + 0.0713i)T \)
31 \( 1 + (-0.540 - 0.841i)T \)
37 \( 1 + (0.959 - 0.281i)T \)
41 \( 1 + (0.977 + 0.212i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.755 + 0.654i)T \)
53 \( 1 + (-0.349 + 0.936i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.599 + 0.800i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.0713 - 0.997i)T \)
79 \( 1 + (0.909 + 0.415i)T \)
83 \( 1 + (0.281 - 0.959i)T \)
89 \( 1 + (0.479 + 0.877i)T \)
97 \( 1 + (0.479 + 0.877i)T \)
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

−17.67611232711529801446425614405, −16.61660889098410191192448985193, −16.006899770294170969498734163092, −15.83866278105991928712786639270, −15.0356788021182261765590566310, −14.184497831739923515772603887184, −13.30513900774681645319630870981, −12.96481796310675935579197819400, −12.28011026833940228199846156768, −11.51683270864564619129939477380, −11.29163231417047444129050215573, −10.884021508718055130768602084759, −9.622653087808052776139509382937, −8.97081953695297952203855206827, −8.07403588729599632605659372561, −7.33355048219200097587843253224, −6.52715285774317706893164315270, −5.894228293050388208097944531014, −5.31917650541084060719946919629, −4.73980341496062776531632842743, −3.86253281785594474033239642114, −3.22584333833212830775753629908, −2.249385433054150905560823087652, −1.41483406273095222249504704682, −0.61423495652423775749732218157, 0.65750247840790949905839139590, 1.82616827544349266716846114907, 3.00404407371520321201709120470, 3.80521095409283624209853746502, 4.18538960881063017429795964601, 4.71832722391387112900258841232, 5.731788578595462963878549907326, 6.23585065763619243339327595426, 7.01654950839279701419969166319, 7.656652735598487184154191838957, 7.987424059842593436179332129597, 9.29756474889673175673451969068, 10.240306934483824422241581840800, 10.85562578018686708397079353101, 11.15300749019417462185775049785, 12.02117799099209332053380084586, 12.66565285688737256192477015065, 13.02295267280107327456174086487, 14.083839507951589993146387231739, 14.807036063857202332372572507143, 15.123168326625711741337787995984, 16.03461406717528663331209761277, 16.31561203584684891615143555437, 16.93188698176878832601193525904, 17.764776961973326649031700298490

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