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Properties

Label 1-6001-6001.349-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.919 + 0.391i$
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  + (−0.909 − 0.415i)2-s + (−0.936 − 0.349i)3-s + (0.654 + 0.755i)4-s + (0.936 + 0.349i)5-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s + (−0.281 − 0.959i)8-s + (0.755 + 0.654i)9-s + (−0.707 − 0.707i)10-s + (0.212 − 0.977i)11-s + (−0.349 − 0.936i)12-s + (0.989 + 0.142i)13-s + (−0.349 − 0.936i)14-s + (−0.755 − 0.654i)15-s + (−0.142 + 0.989i)16-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)2-s + (−0.936 − 0.349i)3-s + (0.654 + 0.755i)4-s + (0.936 + 0.349i)5-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s + (−0.281 − 0.959i)8-s + (0.755 + 0.654i)9-s + (−0.707 − 0.707i)10-s + (0.212 − 0.977i)11-s + (−0.349 − 0.936i)12-s + (0.989 + 0.142i)13-s + (−0.349 − 0.936i)14-s + (−0.755 − 0.654i)15-s + (−0.142 + 0.989i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.919 + 0.391i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.919 + 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.237018883 + 0.2525625292i\)
\(L(\frac12)\) \(\approx\) \(1.237018883 + 0.2525625292i\)
\(L(1)\) \(\approx\) \(0.7802322239 - 0.05115698693i\)
\(L(1)\) \(\approx\) \(0.7802322239 - 0.05115698693i\)

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 \( 1 + (-0.936 - 0.349i)T \)
5 \( 1 + (0.936 + 0.349i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.212 - 0.977i)T \)
13 \( 1 + (0.989 + 0.142i)T \)
19 \( 1 + (0.989 - 0.142i)T \)
23 \( 1 + (-0.977 + 0.212i)T \)
29 \( 1 + (0.997 - 0.0713i)T \)
31 \( 1 + (-0.212 + 0.977i)T \)
37 \( 1 + (-0.479 + 0.877i)T \)
41 \( 1 + (0.212 - 0.977i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.599 - 0.800i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.997 - 0.0713i)T \)
73 \( 1 + (-0.0713 + 0.997i)T \)
79 \( 1 + (-0.936 + 0.349i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
89 \( 1 + (-0.281 + 0.959i)T \)
97 \( 1 + (0.479 + 0.877i)T \)
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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

−17.70372575047915155134207190652, −17.16674387448677371941478953718, −16.34874222815029199365048101748, −16.14691940120672518095167471749, −15.23023444113715763042441715902, −14.497623289423071039346418305982, −13.909851705125536398035762424787, −13.07660625435026065301413701068, −12.19679341042416104874888310322, −11.50901516718091529246445382045, −10.915374920155625534486928290075, −10.145046353436611902723407581869, −9.88939857587235468251619363196, −9.16143402042095082530358732546, −8.32697696959635048099415863579, −7.53999242696421452847903657282, −6.921712283297862371926576953258, −5.977507602712040231744473929582, −5.848536105308995946325339368848, −4.69281425628893286310568668718, −4.45143683339358266195984373709, −3.10046416692352993568923431526, −1.730081317227859682022427168436, −1.48685290829079428230125089640, −0.59010994830482770900358570997, 0.93283514958529192193324499995, 1.49487885715695438883172997309, 2.1427326260738309361687798145, 3.0495567858573202750103535047, 3.8637142892358074721705373719, 5.174102289666848068191049925118, 5.604058013799976742489637501795, 6.44933806989589196646129792573, 6.79517715577101018737216250200, 7.84076703616060554533056024532, 8.52013007100505024251361166026, 9.024259111748193075078602004499, 10.053091672597694931246919658107, 10.405768426641576525889731402962, 11.24725563915962146328482642939, 11.6084298597624123354300916288, 12.167229664509505276933481825861, 13.07285607342987095733352171174, 13.74137211685103772935799985626, 14.27866039376985296877125519481, 15.55249512522920500939726473929, 15.99095623501771736348223542881, 16.58376269597309632024041390200, 17.56120532419989941206336461986, 17.63942893371304003640818212725

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