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Properties

Label 1-6001-6001.2435-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.997 + 0.0708i$
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  + i·2-s + (0.382 + 0.923i)3-s − 4-s + (0.923 − 0.382i)5-s + (−0.923 + 0.382i)6-s + (−0.923 − 0.382i)7-s − i·8-s + (−0.707 + 0.707i)9-s + (0.382 + 0.923i)10-s − 11-s + (−0.382 − 0.923i)12-s + (−0.923 + 0.382i)13-s + (0.382 − 0.923i)14-s + (0.707 + 0.707i)15-s + 16-s + ⋯
L(s)  = 1  + i·2-s + (0.382 + 0.923i)3-s − 4-s + (0.923 − 0.382i)5-s + (−0.923 + 0.382i)6-s + (−0.923 − 0.382i)7-s − i·8-s + (−0.707 + 0.707i)9-s + (0.382 + 0.923i)10-s − 11-s + (−0.382 − 0.923i)12-s + (−0.923 + 0.382i)13-s + (0.382 − 0.923i)14-s + (0.707 + 0.707i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.997 + 0.0708i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (2435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.997 + 0.0708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6879992065 + 0.02440596956i\)
\(L(\frac12)\) \(\approx\) \(0.6879992065 + 0.02440596956i\)
\(L(1)\) \(\approx\) \(0.6961653679 + 0.5495514651i\)
\(L(1)\) \(\approx\) \(0.6961653679 + 0.5495514651i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (-0.923 - 0.382i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.382 - 0.923i)T \)
59 \( 1 + (-0.382 + 0.923i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.923 - 0.382i)T \)
97 \( 1 - iT \)
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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

−17.89015239322409282597209679257, −17.4287598442020029332313419867, −16.71270427725263008512038264952, −15.463557149847072494611078022262, −14.998774016543084387989842887123, −14.02801807764248550558455350383, −13.55184427018508758344800144246, −13.16367843939236770542944599790, −12.44487733212242670947047690094, −12.04720688928855344736220340408, −11.07301616198838777844312997576, −10.352610044697657882963335478643, −9.77885011853721462770081320003, −9.139946368863620348480284813871, −8.63841000507016959217644347894, −7.52041261830276730876196677033, −7.11765903383207957724539013835, −6.10800772296045603976766254761, −5.42819875647443680235371181449, −4.926464239156524040859584299135, −3.362131480956804355908246977561, −3.070126433495647071552281938244, −2.41272941173650659502291856728, −1.86379840895429122167657564522, −0.81004176537338504795979838953, 0.19386985481539198814451088231, 1.54979125930045706397005061271, 2.82153451072750882726910786629, 3.18487607865102354125522112461, 4.38435707264596597350715081352, 4.78748644185751117653143650844, 5.52440254305677745679412295778, 6.06617271138038811455543548155, 6.996502695678460915560646200835, 7.59676695515626503944503956484, 8.514861574905322060345389779032, 9.05435719136643599027678518482, 9.69401935458847937291111674511, 10.20428158032743964626545344409, 10.56530591658621387554839364134, 11.983042946545074967611145809818, 12.89459541603674784364604970993, 13.18481633322113627355542486183, 14.12152093726865468943460778830, 14.34900220243385834487518870667, 15.154935804662345437299395677, 16.04054670661016107082403614090, 16.28492712813403981137649481898, 16.80196488827781014780240586965, 17.50656055861883110439105429393

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