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Properties

Label 1-3021-3021.344-r1-0-0
Degree $1$
Conductor $3021$
Sign $0.929 + 0.367i$
Analytic cond. $324.651$
Root an. cond. $324.651$
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  + (0.974 + 0.226i)2-s + (0.897 + 0.440i)4-s + (0.995 − 0.0938i)5-s + (0.919 + 0.391i)7-s + (0.774 + 0.632i)8-s + (0.991 + 0.133i)10-s + (0.948 + 0.316i)11-s + (0.404 − 0.914i)13-s + (0.807 + 0.589i)14-s + (0.611 + 0.791i)16-s + (0.653 − 0.757i)17-s + (0.935 + 0.354i)20-s + (0.852 + 0.523i)22-s + (−0.642 + 0.766i)23-s + (0.982 − 0.186i)25-s + (0.600 − 0.799i)26-s + ⋯
L(s)  = 1  + (0.974 + 0.226i)2-s + (0.897 + 0.440i)4-s + (0.995 − 0.0938i)5-s + (0.919 + 0.391i)7-s + (0.774 + 0.632i)8-s + (0.991 + 0.133i)10-s + (0.948 + 0.316i)11-s + (0.404 − 0.914i)13-s + (0.807 + 0.589i)14-s + (0.611 + 0.791i)16-s + (0.653 − 0.757i)17-s + (0.935 + 0.354i)20-s + (0.852 + 0.523i)22-s + (−0.642 + 0.766i)23-s + (0.982 − 0.186i)25-s + (0.600 − 0.799i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3021\)    =    \(3 \cdot 19 \cdot 53\)
Sign: $0.929 + 0.367i$
Analytic conductor: \(324.651\)
Root analytic conductor: \(324.651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3021} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3021,\ (1:\ ),\ 0.929 + 0.367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(8.652144986 + 1.649072831i\)
\(L(\frac12)\) \(\approx\) \(8.652144986 + 1.649072831i\)
\(L(1)\) \(\approx\) \(2.988185275 + 0.4623729956i\)
\(L(1)\) \(\approx\) \(2.988185275 + 0.4623729956i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (0.974 + 0.226i)T \)
5 \( 1 + (0.995 - 0.0938i)T \)
7 \( 1 + (0.919 + 0.391i)T \)
11 \( 1 + (0.948 + 0.316i)T \)
13 \( 1 + (0.404 - 0.914i)T \)
17 \( 1 + (0.653 - 0.757i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.930 - 0.367i)T \)
31 \( 1 + (-0.316 - 0.948i)T \)
37 \( 1 + (0.885 - 0.464i)T \)
41 \( 1 + (0.621 - 0.783i)T \)
43 \( 1 + (-0.977 + 0.213i)T \)
47 \( 1 + (0.815 + 0.579i)T \)
59 \( 1 + (0.252 + 0.967i)T \)
61 \( 1 + (-0.944 - 0.329i)T \)
67 \( 1 + (0.556 + 0.830i)T \)
71 \( 1 + (-0.213 - 0.977i)T \)
73 \( 1 + (0.186 - 0.982i)T \)
79 \( 1 + (-0.0536 - 0.998i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.859 + 0.511i)T \)
97 \( 1 + (0.815 - 0.579i)T \)
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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

−18.75051690292513893601069233611, −18.325919558352168321033753113492, −17.155611422466960710590804976616, −16.79879062221013903946741114082, −16.118881003461791202683670352695, −14.8368238276707117119268448455, −14.51709896895309402619892368438, −14.006083602928471637415568204102, −13.368286482643795893942757837226, −12.55100146968751032038804705071, −11.78398859999809941093022331006, −11.11492181572249047628210591479, −10.51628979494613362659138365331, −9.74031896833048680802001502405, −8.87342817902271440880021558952, −7.98123613460247457601352488528, −6.92499251914559394658691500251, −6.38358972981718939727756412788, −5.69004047029861122342308987359, −4.89660897822547833086226907152, −4.09884748237830864779307563364, −3.468589067987074566982418479300, −2.2973319020153307048829833146, −1.58749879031389509624647770059, −1.09642987640665733897826558024, 0.99295593610806463802728021135, 1.82180030310808639497811852685, 2.476066423315922524731714794153, 3.48079185533346355223645186773, 4.31293012472946869642026308482, 5.18006248742783580537299556122, 5.765981581310682773948498739, 6.21954411175616667309316573421, 7.43082607980976912606065992154, 7.82373703967843047751733437326, 8.94515963139606968696188030231, 9.608154724536674255212176160660, 10.57104108898329889407016314405, 11.31643168585970595462106553582, 11.961009703656050817097595674920, 12.63700601612481963343696702171, 13.508223727294588762215885922812, 13.91866118286324967943205793350, 14.80656949663086064668170720434, 15.03243272240708354074697126959, 16.09972207874950967042946587586, 16.789998785818832980736900994090, 17.50055685771455633132258310069, 17.970510327208090697893729032814, 18.84724484790899069457740608939

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