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Properties

Label 1-6001-6001.4807-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.316 - 0.948i$
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  + (0.654 − 0.755i)2-s + (−0.654 − 0.755i)3-s + (−0.142 − 0.989i)4-s + (−0.654 − 0.755i)5-s − 6-s + 7-s + (−0.841 − 0.540i)8-s + (−0.142 + 0.989i)9-s − 10-s + (0.909 − 0.415i)11-s + (−0.654 + 0.755i)12-s + (0.281 + 0.959i)13-s + (0.654 − 0.755i)14-s + (−0.142 + 0.989i)15-s + (−0.959 + 0.281i)16-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (−0.654 − 0.755i)3-s + (−0.142 − 0.989i)4-s + (−0.654 − 0.755i)5-s − 6-s + 7-s + (−0.841 − 0.540i)8-s + (−0.142 + 0.989i)9-s − 10-s + (0.909 − 0.415i)11-s + (−0.654 + 0.755i)12-s + (0.281 + 0.959i)13-s + (0.654 − 0.755i)14-s + (−0.142 + 0.989i)15-s + (−0.959 + 0.281i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (4807, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.316 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.321333432 - 0.9523708211i\)
\(L(\frac12)\) \(\approx\) \(1.321333432 - 0.9523708211i\)
\(L(1)\) \(\approx\) \(0.8623627955 - 0.7440588551i\)
\(L(1)\) \(\approx\) \(0.8623627955 - 0.7440588551i\)

Euler product

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

−17.45266686448550268145266893930, −17.16953033127853526516512093541, −16.5916188500180643562052822221, −15.504022542695418406873092413, −15.2918262741084877877295144670, −14.74982267892534202299998051438, −14.32013951913915833006197223932, −13.33142570370731790074910296638, −12.4423661136031267255357656156, −11.900334893373130905546368613779, −11.31816980303710487552584250702, −10.74954054420338873515003662986, −10.06048225530600894374830793757, −8.9199586129212590922664058755, −8.44576228126359405630252510324, −7.6307673027025224913200845043, −6.90244501731175335672280779133, −6.37725599993205970054373203857, −5.52699263645036238670365390384, −4.954857913176290004293813470115, −4.13696799626316368640855671959, −3.7872224276171513437293166075, −2.982918658660029867532538243804, −1.89639434573193391310074793599, −0.440764988551847147085513149648, 0.8590508257290085055921276006, 1.47482112824286628031595511399, 1.92606589790405371102678825885, 3.11638194242241030721896652180, 4.08580369641840287782822928928, 4.562956798065476605743888077335, 5.18134623726898385428953190252, 5.92606043971255633132143148592, 6.70489025814209104256226427786, 7.329210943521570152854380529223, 8.40503554884149842932755025825, 8.804072927717947568262985476123, 9.61695669549164792847518381920, 10.89057397967971079892176028633, 11.23243772843716127705481881823, 11.536410810895459953928266461412, 12.41400053239712528075111561781, 12.67169296955954456630632082275, 13.63410033821459954610286645696, 14.0495923713465773016400189183, 14.79716167740812121066037460671, 15.54425963243932933118149310984, 16.356176867096872536254505961329, 17.090763144788097463776790830345, 17.48383126843157960642183727274

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