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Properties

Label 1-6027-6027.1076-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.996 + 0.0893i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (0.280 − 0.959i)2-s + (−0.842 − 0.538i)4-s + (−0.599 + 0.800i)5-s + (−0.753 + 0.657i)8-s + (0.599 + 0.800i)10-s + (0.646 − 0.762i)11-s + (0.691 + 0.722i)13-s + (0.420 + 0.907i)16-s + (0.887 + 0.460i)17-s + (0.978 − 0.207i)19-s + (0.936 − 0.351i)20-s + (−0.550 − 0.834i)22-s + (0.163 + 0.986i)23-s + (−0.280 − 0.959i)25-s + (0.887 − 0.460i)26-s + ⋯
L(s)  = 1  + (0.280 − 0.959i)2-s + (−0.842 − 0.538i)4-s + (−0.599 + 0.800i)5-s + (−0.753 + 0.657i)8-s + (0.599 + 0.800i)10-s + (0.646 − 0.762i)11-s + (0.691 + 0.722i)13-s + (0.420 + 0.907i)16-s + (0.887 + 0.460i)17-s + (0.978 − 0.207i)19-s + (0.936 − 0.351i)20-s + (−0.550 − 0.834i)22-s + (0.163 + 0.986i)23-s + (−0.280 − 0.959i)25-s + (0.887 − 0.460i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.996 + 0.0893i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1076, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.996 + 0.0893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.863074137 + 0.08339111316i\)
\(L(\frac12)\) \(\approx\) \(1.863074137 + 0.08339111316i\)
\(L(1)\) \(\approx\) \(1.112534733 - 0.2887668065i\)
\(L(1)\) \(\approx\) \(1.112534733 - 0.2887668065i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.280 - 0.959i)T \)
5 \( 1 + (-0.599 + 0.800i)T \)
11 \( 1 + (0.646 - 0.762i)T \)
13 \( 1 + (0.691 + 0.722i)T \)
17 \( 1 + (0.887 + 0.460i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (0.163 + 0.986i)T \)
29 \( 1 + (0.963 + 0.266i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.251 + 0.967i)T \)
43 \( 1 + (0.858 - 0.512i)T \)
47 \( 1 + (-0.280 + 0.959i)T \)
53 \( 1 + (0.842 + 0.538i)T \)
59 \( 1 + (0.0149 + 0.999i)T \)
61 \( 1 + (0.772 + 0.635i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (0.963 - 0.266i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.971 + 0.237i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−17.564759723160515152262275145664, −16.8948869891490923251552048167, −16.158153815877069122685747575205, −15.979187847925995114261067671173, −15.10721556782061070954356215549, −14.509232658922917906468110838781, −13.94710303651399217940045187531, −12.98586273019494388777401877704, −12.641079639609742842494669335912, −11.96887035224502626613409247802, −11.343847856251589922072723043086, −10.15690105371601486914804502134, −9.54234927631383216241653119457, −8.85124867347297212031907849605, −8.22091977399565415471796275619, −7.608988145084951586560840985081, −7.04195524599406911060382972896, −6.16902090110130974008467214658, −5.37221233230224017534514139730, −4.95338280674974656997873145657, −3.98111885310939133325507280805, −3.66155156936025684386046398861, −2.61495538384689773690571627516, −1.225336884920487758782812870892, −0.56527982600165689808374503505, 0.993326835346838331122157384787, 1.480485032737582375964862946709, 2.67965938676239688688208434417, 3.3160678952441699902955867983, 3.751982284730413789752158828751, 4.47403806231846183218729764484, 5.51130860200113396562623014885, 6.07019867984285432684758915853, 6.89816071059114739682724082381, 7.71881403293187249840244508802, 8.54165946335109227818520826477, 9.15239624231252121110889277772, 9.88652005590561505285807435903, 10.674605907695891684939362129648, 11.13108756375196484105925185947, 11.88528002582369284774054287740, 12.04502703985043913209655146640, 13.194085897387505485748217442782, 13.818177940715817598805512453748, 14.29614901975242263020103674941, 14.8454095720301144703033899760, 15.73797914145260768015658857751, 16.29605097013423776270294102088, 17.21980120864452829782929334022, 17.97184943656452857884331619547

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