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Properties

Label 1-6001-6001.36-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.829 - 0.558i$
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  + 2-s + (0.382 − 0.923i)3-s + 4-s + (0.923 + 0.382i)5-s + (0.382 − 0.923i)6-s + (−0.923 + 0.382i)7-s + 8-s + (−0.707 − 0.707i)9-s + (0.923 + 0.382i)10-s + (−0.707 + 0.707i)11-s + (0.382 − 0.923i)12-s + (−0.923 + 0.382i)13-s + (−0.923 + 0.382i)14-s + (0.707 − 0.707i)15-s + 16-s + ⋯
L(s)  = 1  + 2-s + (0.382 − 0.923i)3-s + 4-s + (0.923 + 0.382i)5-s + (0.382 − 0.923i)6-s + (−0.923 + 0.382i)7-s + 8-s + (−0.707 − 0.707i)9-s + (0.923 + 0.382i)10-s + (−0.707 + 0.707i)11-s + (0.382 − 0.923i)12-s + (−0.923 + 0.382i)13-s + (−0.923 + 0.382i)14-s + (0.707 − 0.707i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.829 - 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.202781331 - 1.282291476i\)
\(L(\frac12)\) \(\approx\) \(4.202781331 - 1.282291476i\)
\(L(1)\) \(\approx\) \(2.248121520 - 0.4375006864i\)
\(L(1)\) \(\approx\) \(2.248121520 - 0.4375006864i\)

Euler product

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

−17.45754858068744304595146350145, −16.76115941571867841314325369210, −16.38805531490789468851357152345, −15.79302425511432463964176590635, −15.147605696213871831213581257503, −14.35111029755870105436195402896, −13.87147242152492543511169392958, −13.25647233485394974238508953862, −12.82884056980781491976567722927, −11.9505528356258168606529666120, −11.10004741366976467252438785279, −10.40011356998450533211471699798, −9.85539427276229637346567598309, −9.47079053390526505660181092445, −8.37792671439279568351910730870, −7.66545068089848307314627960530, −6.87646225258773808524765364441, −5.83740648678670361576741821202, −5.51603717756115740980776620859, −4.95458218782664387872332646478, −3.98052444832352090538533250484, −3.38273499359839042581061561556, −2.726680065845044278992544951860, −2.12340413404419385638409864051, −0.871033905916829421312093445524, 0.814882594580965901857536485470, 2.09019576232352169985518396832, 2.396031148532900563905371376024, 2.87711585122133526921176661290, 3.7663570020979926400749124636, 4.92453735248914957094698517748, 5.422781711653022522550899606701, 6.28865196312899327037999547428, 6.778056684629267484753621683827, 7.206043622016626628725508194879, 8.049611042823069634262919577645, 9.05226754622024038126221663042, 9.84065482371795818084685986064, 10.233425010798196287856397509946, 11.32344151239222970025775115658, 12.02946072038041902597214490021, 12.57900891830290399401514809769, 13.11845063181724959802429945486, 13.65631384406501132194821300825, 14.15645923122158115690981299380, 15.05663397700641104396544911308, 15.2632984895453294059549073524, 16.28987119525597618796113130912, 17.06000598643661093973043751523, 17.58023620751119424642860542422

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