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Properties

Label 1-3021-3021.14-r1-0-0
Degree $1$
Conductor $3021$
Sign $0.838 + 0.544i$
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  + (0.440 + 0.897i)2-s + (−0.611 + 0.791i)4-s + (−0.186 + 0.982i)5-s + (−0.692 + 0.721i)7-s + (−0.979 − 0.200i)8-s + (−0.964 + 0.265i)10-s + (0.799 − 0.600i)11-s + (0.673 − 0.739i)13-s + (−0.952 − 0.303i)14-s + (−0.252 − 0.967i)16-s + (−0.147 + 0.989i)17-s + (−0.663 − 0.748i)20-s + (0.891 + 0.452i)22-s + (−0.984 − 0.173i)23-s + (−0.930 − 0.367i)25-s + (0.960 + 0.278i)26-s + ⋯
L(s)  = 1  + (0.440 + 0.897i)2-s + (−0.611 + 0.791i)4-s + (−0.186 + 0.982i)5-s + (−0.692 + 0.721i)7-s + (−0.979 − 0.200i)8-s + (−0.964 + 0.265i)10-s + (0.799 − 0.600i)11-s + (0.673 − 0.739i)13-s + (−0.952 − 0.303i)14-s + (−0.252 − 0.967i)16-s + (−0.147 + 0.989i)17-s + (−0.663 − 0.748i)20-s + (0.891 + 0.452i)22-s + (−0.984 − 0.173i)23-s + (−0.930 − 0.367i)25-s + (0.960 + 0.278i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3021\)    =    \(3 \cdot 19 \cdot 53\)
Sign: $0.838 + 0.544i$
Analytic conductor: \(324.651\)
Root analytic conductor: \(324.651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3021} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3021,\ (1:\ ),\ 0.838 + 0.544i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7420346608 + 0.2197124384i\)
\(L(\frac12)\) \(\approx\) \(0.7420346608 + 0.2197124384i\)
\(L(1)\) \(\approx\) \(0.6931889923 + 0.6607664707i\)
\(L(1)\) \(\approx\) \(0.6931889923 + 0.6607664707i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (0.440 + 0.897i)T \)
5 \( 1 + (-0.186 + 0.982i)T \)
7 \( 1 + (-0.692 + 0.721i)T \)
11 \( 1 + (0.799 - 0.600i)T \)
13 \( 1 + (0.673 - 0.739i)T \)
17 \( 1 + (-0.147 + 0.989i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (-0.730 + 0.682i)T \)
31 \( 1 + (-0.600 + 0.799i)T \)
37 \( 1 + (0.568 + 0.822i)T \)
41 \( 1 + (-0.974 - 0.226i)T \)
43 \( 1 + (-0.909 - 0.416i)T \)
47 \( 1 + (-0.329 + 0.944i)T \)
59 \( 1 + (0.872 + 0.488i)T \)
61 \( 1 + (-0.621 - 0.783i)T \)
67 \( 1 + (0.925 - 0.379i)T \)
71 \( 1 + (-0.416 + 0.909i)T \)
73 \( 1 + (-0.367 - 0.930i)T \)
79 \( 1 + (-0.107 + 0.994i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.476 - 0.879i)T \)
97 \( 1 + (-0.329 - 0.944i)T \)
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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

−19.02639140978100062304888834281, −18.280286699802665917781429983068, −17.46080073781133926736117241644, −16.61515441441401516731083344321, −16.162220193136662345340253011162, −15.23054805630450494065529586885, −14.3991749117618129848594163565, −13.528978648089982335317331401990, −13.30075125857712199706281229152, −12.44523866008028405870646545356, −11.66467998012955148861545872625, −11.37486176014716022183161184259, −10.17538936513996867489946683563, −9.527914704330300346000978885844, −9.18674143326091880308952542645, −8.208604057083057009482134243879, −7.1653007777367725758973652533, −6.343271892781395276877604950588, −5.53752276043523092379766371634, −4.58221496299150562650757215665, −4.00176518881281044245744019227, −3.55044767750565082366075503195, −2.216687122215181842954829286, −1.508907851714321637113190198423, −0.620937383178072899695384697063, 0.15206772186291812591783331153, 1.721640758663987914797093539395, 3.000526834446489273718230679612, 3.43623803839564729373254665763, 4.07825058151039872833487795184, 5.36601662943255603209282424536, 6.05228386003313237730694952033, 6.44736787612998496600460624308, 7.17946990885857212970407301903, 8.24363924530755433650177790169, 8.59307827063889965678873668513, 9.563429670025052302632918928252, 10.369291713007498159209104604364, 11.274634022048415148857512630217, 12.00122897441301779188221864361, 12.761457171326669927956947440576, 13.40636749377261437922554293025, 14.22994489348179910714020642032, 14.79908385110313133255225793300, 15.42040116993299821689051059202, 15.9853870352562449175192907360, 16.70122687774616215130606266943, 17.486631694438154165457267097802, 18.36235258082400859427407538717, 18.604201929773496344972958818828

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