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Properties

Label 1-2028-2028.287-r0-0-0
Degree $1$
Conductor $2028$
Sign $0.867 + 0.497i$
Analytic cond. $9.41799$
Root an. cond. $9.41799$
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  + (−0.885 − 0.464i)5-s + (0.354 + 0.935i)7-s + (0.120 − 0.992i)11-s + (0.354 + 0.935i)17-s − 19-s + 23-s + (0.568 + 0.822i)25-s + (−0.120 − 0.992i)29-s + (−0.568 + 0.822i)31-s + (0.120 − 0.992i)35-s + (0.568 − 0.822i)37-s + (0.748 + 0.663i)41-s + (−0.568 − 0.822i)43-s + (−0.970 − 0.239i)47-s + (−0.748 + 0.663i)49-s + ⋯
L(s)  = 1  + (−0.885 − 0.464i)5-s + (0.354 + 0.935i)7-s + (0.120 − 0.992i)11-s + (0.354 + 0.935i)17-s − 19-s + 23-s + (0.568 + 0.822i)25-s + (−0.120 − 0.992i)29-s + (−0.568 + 0.822i)31-s + (0.120 − 0.992i)35-s + (0.568 − 0.822i)37-s + (0.748 + 0.663i)41-s + (−0.568 − 0.822i)43-s + (−0.970 − 0.239i)47-s + (−0.748 + 0.663i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.867 + 0.497i$
Analytic conductor: \(9.41799\)
Root analytic conductor: \(9.41799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2028,\ (0:\ ),\ 0.867 + 0.497i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217821628 + 0.3242931736i\)
\(L(\frac12)\) \(\approx\) \(1.217821628 + 0.3242931736i\)
\(L(1)\) \(\approx\) \(0.9488865425 + 0.04102620276i\)
\(L(1)\) \(\approx\) \(0.9488865425 + 0.04102620276i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.885 - 0.464i)T \)
7 \( 1 + (0.354 + 0.935i)T \)
11 \( 1 + (0.120 - 0.992i)T \)
17 \( 1 + (0.354 + 0.935i)T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + (-0.120 - 0.992i)T \)
31 \( 1 + (-0.568 + 0.822i)T \)
37 \( 1 + (0.568 - 0.822i)T \)
41 \( 1 + (0.748 + 0.663i)T \)
43 \( 1 + (-0.568 - 0.822i)T \)
47 \( 1 + (-0.970 - 0.239i)T \)
53 \( 1 + (0.354 + 0.935i)T \)
59 \( 1 + (0.885 + 0.464i)T \)
61 \( 1 + (-0.354 + 0.935i)T \)
67 \( 1 + (0.970 + 0.239i)T \)
71 \( 1 + (-0.748 - 0.663i)T \)
73 \( 1 + (0.120 - 0.992i)T \)
79 \( 1 + (0.970 + 0.239i)T \)
83 \( 1 + (-0.748 + 0.663i)T \)
89 \( 1 - T \)
97 \( 1 + (0.885 - 0.464i)T \)
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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

−19.93513352967605844083784904137, −19.17388663161338334600463493673, −18.43390738390868778564655828342, −17.71870428188250346158167173318, −16.89777113910767130761433162153, −16.30450964551627014951835575817, −15.374460980687247509385037417680, −14.67351095607484744523569957200, −14.30395126710556029934459184767, −13.09930286886656477419494090858, −12.61015168837516637565310012120, −11.47309878523912529370031942313, −11.16971197006182114924805612631, −10.2525493904683927177807900762, −9.56015226122860572899593536032, −8.483313147215891495175727760703, −7.698180376267656489336552602789, −7.08826878799121303462027051913, −6.561570355190877527245098176656, −5.10857902616681277374358192902, −4.494708314206233393534159238421, −3.7358733657628778416347774892, −2.86459630776084568803731654946, −1.75988314727652279146351058923, −0.59104398512292341672555410945, 0.83500021447381562231255838851, 1.92209699571666300789033238601, 2.99599318245553804462007787493, 3.82247819838629063639014663654, 4.646772273605417956708844474357, 5.55794688302014131880737917625, 6.1952671730905183433963507466, 7.29536842512131710533743627983, 8.22987690081946361041034600200, 8.60644752819847080488664287016, 9.29200757981617161740145086690, 10.55355904918696340410766534423, 11.20962285493510327309297231698, 11.83056172517540753684906320807, 12.64414236180777785611454219380, 13.13931752329622516077804969233, 14.318645686753488333881193561285, 15.030372883637831457517527825235, 15.460711373154111570309855511942, 16.48642487544333187568475686377, 16.8326985799433783677593265204, 17.8722904631519027149138922902, 18.68798204371570990451954090335, 19.37053793409085169966638327558, 19.649510739727868908818207075466

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