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Properties

Label 1-6001-6001.191-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.294 - 0.955i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.142 − 0.989i)3-s + (−0.959 − 0.281i)4-s + (−0.142 − 0.989i)5-s − 6-s + 7-s + (−0.415 + 0.909i)8-s + (−0.959 + 0.281i)9-s − 10-s + (0.755 − 0.654i)11-s + (−0.142 + 0.989i)12-s + (−0.540 + 0.841i)13-s + (0.142 − 0.989i)14-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.142 − 0.989i)3-s + (−0.959 − 0.281i)4-s + (−0.142 − 0.989i)5-s − 6-s + 7-s + (−0.415 + 0.909i)8-s + (−0.959 + 0.281i)9-s − 10-s + (0.755 − 0.654i)11-s + (−0.142 + 0.989i)12-s + (−0.540 + 0.841i)13-s + (0.142 − 0.989i)14-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.294 - 0.955i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.294 - 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9800862528 - 1.328181955i\)
\(L(\frac12)\) \(\approx\) \(0.9800862528 - 1.328181955i\)
\(L(1)\) \(\approx\) \(0.6915433087 - 0.8160205993i\)
\(L(1)\) \(\approx\) \(0.6915433087 - 0.8160205993i\)

Euler product

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

−17.92124610754608331557486688943, −17.073338955524768692024162959981, −16.73408777349603339701427638956, −15.67021689865683255807577078795, −15.2232111352312336722399925696, −14.847443513096862046683446988633, −14.14192193149137661202381633827, −13.92692399810421877955243715014, −12.541698120812313602603790277447, −11.99703627301791896950749932249, −11.232598131295714832671667654384, −10.449561823050510223057463038873, −9.97098005116825405705496278595, −9.21157782524424589472330104317, −8.4934454664457520437042893046, −7.73192579149520777195511634178, −7.20001879028889367988832146906, −6.41716915640191171730191599470, −5.602955199161783814512878390122, −5.03865242239511539759478110287, −4.39297900921231924562681378708, −3.63744604639508796007512846591, −3.04910740557944729732733966742, −1.94269092367945952002125240842, −0.513633943623463097442214102680, 0.91299169124768242888645407069, 1.2671907554870800683659480430, 1.942226789609051486153597140598, 2.82643056814114070740842250429, 3.749740510109110338476381157898, 4.544944792113082998099669504368, 5.274439028915990121649044325756, 5.62175905198840905296848573831, 6.84563379642582375960463956082, 7.50827432355150467009196995147, 8.40306034445656228302682069623, 8.855951350775998279850811127513, 9.32418212979212172765305973919, 10.39842219801060650544303745735, 11.40305485397378900788529178885, 11.65375836475206889477950442754, 11.99294997868529510126083190419, 12.89947025657758904993613774176, 13.47855501154979325668705183100, 13.98142623402417366910312050829, 14.5574890868041277330192180233, 15.39847787415413415428159999362, 16.71659136752953777038973977906, 16.95189979369037828777891197115, 17.56986305954950816168396896427

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