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Properties

Label 1-6011-6011.12-r0-0-0
Degree $1$
Conductor $6011$
Sign $0.996 - 0.0782i$
Analytic cond. $27.9149$
Root an. cond. $27.9149$
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  + (−0.0371 − 0.999i)2-s + (0.821 + 0.569i)3-s + (−0.997 + 0.0741i)4-s + (0.998 − 0.0564i)5-s + (0.539 − 0.842i)6-s + (−0.741 + 0.670i)7-s + (0.111 + 0.993i)8-s + (0.350 + 0.936i)9-s + (−0.0934 − 0.995i)10-s + (0.958 + 0.284i)11-s + (−0.861 − 0.507i)12-s + (−0.178 − 0.983i)13-s + (0.697 + 0.716i)14-s + (0.852 + 0.522i)15-s + (0.989 − 0.147i)16-s + (−0.989 − 0.144i)17-s + ⋯
L(s)  = 1  + (−0.0371 − 0.999i)2-s + (0.821 + 0.569i)3-s + (−0.997 + 0.0741i)4-s + (0.998 − 0.0564i)5-s + (0.539 − 0.842i)6-s + (−0.741 + 0.670i)7-s + (0.111 + 0.993i)8-s + (0.350 + 0.936i)9-s + (−0.0934 − 0.995i)10-s + (0.958 + 0.284i)11-s + (−0.861 − 0.507i)12-s + (−0.178 − 0.983i)13-s + (0.697 + 0.716i)14-s + (0.852 + 0.522i)15-s + (0.989 − 0.147i)16-s + (−0.989 − 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6011\)
Sign: $0.996 - 0.0782i$
Analytic conductor: \(27.9149\)
Root analytic conductor: \(27.9149\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6011} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6011,\ (0:\ ),\ 0.996 - 0.0782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.605037787 - 0.1020546638i\)
\(L(\frac12)\) \(\approx\) \(2.605037787 - 0.1020546638i\)
\(L(1)\) \(\approx\) \(1.431890406 - 0.2478470283i\)
\(L(1)\) \(\approx\) \(1.431890406 - 0.2478470283i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6011 \( 1 \)
good2 \( 1 + (-0.0371 - 0.999i)T \)
3 \( 1 + (0.821 + 0.569i)T \)
5 \( 1 + (0.998 - 0.0564i)T \)
7 \( 1 + (-0.741 + 0.670i)T \)
11 \( 1 + (0.958 + 0.284i)T \)
13 \( 1 + (-0.178 - 0.983i)T \)
17 \( 1 + (-0.989 - 0.144i)T \)
19 \( 1 + (0.803 - 0.595i)T \)
23 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (0.197 - 0.980i)T \)
31 \( 1 + (-0.965 + 0.259i)T \)
37 \( 1 + (-0.782 + 0.622i)T \)
41 \( 1 + (0.755 + 0.655i)T \)
43 \( 1 + (-0.486 + 0.873i)T \)
47 \( 1 + (0.201 - 0.979i)T \)
53 \( 1 + (-0.192 - 0.981i)T \)
59 \( 1 + (0.992 + 0.120i)T \)
61 \( 1 + (-0.674 - 0.738i)T \)
67 \( 1 + (0.982 - 0.185i)T \)
71 \( 1 + (0.395 - 0.918i)T \)
73 \( 1 + (0.710 + 0.703i)T \)
79 \( 1 + (0.999 - 0.0229i)T \)
83 \( 1 + (0.948 - 0.316i)T \)
89 \( 1 + (0.799 + 0.600i)T \)
97 \( 1 + (0.786 - 0.616i)T \)
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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

−17.624599926459852995419223548800, −17.05072805965680356192378753613, −16.42887922104057367232386289455, −15.89992932627422925818184549784, −14.865491184338277992047548973133, −14.17424762400557695842032886613, −14.069603938249969083964704161009, −13.433440794455247884786002189423, −12.67931903996302078490478086247, −12.23619555105442856299692353606, −10.8970229135863733085005401263, −10.13612391028632333843010273265, −9.229689556212995865351923985742, −9.19781095980422068562207683480, −8.4752835396011325801461541653, −7.30440654660108922329536075759, −7.01356652414033123546189552444, −6.38215136474038457308603148581, −5.90540304269839253474747334318, −4.79525625694194783045825570482, −3.92512151188453367942774992769, −3.4777076343097910512642995452, −2.34739245919749860016696754108, −1.538733859075702119112000689963, −0.711059006649458588717941665796, 0.8835299763526145875899515204, 2.02440776095260707180790775309, 2.28073872466939169220151845133, 3.320167749818466395618677501017, 3.51721104110043826776684959788, 4.77491176171467675437441177611, 5.1580578430643020960138808531, 6.03674174348089881063509526487, 6.93371851800405726145793856291, 7.96201069261392367964812207907, 8.75749726313958049362641800715, 9.31346400809386789560703563443, 9.657205278951670358242110167707, 10.14897020360645063374325258807, 11.038193511085408224086762616809, 11.698215274094936820874258654079, 12.59247902171857330056611110573, 13.20624311830261923495706723332, 13.566188490014133103921367094834, 14.286648282158036301234257781047, 15.02371559925493193231120176421, 15.55358828368339618472608626459, 16.442351233019716615032685763188, 17.24832269681051047581303396867, 17.85072180302218464245127905117

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