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Properties

Label 1-3887-3887.1019-r0-0-0
Degree $1$
Conductor $3887$
Sign $0.439 + 0.898i$
Analytic cond. $18.0511$
Root an. cond. $18.0511$
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  + (0.217 − 0.975i)2-s + (0.984 − 0.174i)3-s + (−0.905 − 0.425i)4-s + (−0.741 + 0.671i)5-s + (0.0439 − 0.999i)6-s + (−0.869 − 0.493i)7-s + (−0.612 + 0.790i)8-s + (0.938 − 0.344i)9-s + (0.493 + 0.869i)10-s + (−0.217 − 0.975i)11-s + (−0.965 − 0.260i)12-s + (−0.671 + 0.741i)14-s + (−0.612 + 0.790i)15-s + (0.638 + 0.769i)16-s + (−0.980 + 0.196i)17-s + (−0.131 − 0.991i)18-s + ⋯
L(s)  = 1  + (0.217 − 0.975i)2-s + (0.984 − 0.174i)3-s + (−0.905 − 0.425i)4-s + (−0.741 + 0.671i)5-s + (0.0439 − 0.999i)6-s + (−0.869 − 0.493i)7-s + (−0.612 + 0.790i)8-s + (0.938 − 0.344i)9-s + (0.493 + 0.869i)10-s + (−0.217 − 0.975i)11-s + (−0.965 − 0.260i)12-s + (−0.671 + 0.741i)14-s + (−0.612 + 0.790i)15-s + (0.638 + 0.769i)16-s + (−0.980 + 0.196i)17-s + (−0.131 − 0.991i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3887\)    =    \(13^{2} \cdot 23\)
Sign: $0.439 + 0.898i$
Analytic conductor: \(18.0511\)
Root analytic conductor: \(18.0511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3887} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3887,\ (0:\ ),\ 0.439 + 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05345030606 + 0.03335104818i\)
\(L(\frac12)\) \(\approx\) \(0.05345030606 + 0.03335104818i\)
\(L(1)\) \(\approx\) \(0.7644024220 - 0.5450675466i\)
\(L(1)\) \(\approx\) \(0.7644024220 - 0.5450675466i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.217 - 0.975i)T \)
3 \( 1 + (0.984 - 0.174i)T \)
5 \( 1 + (-0.741 + 0.671i)T \)
7 \( 1 + (-0.869 - 0.493i)T \)
11 \( 1 + (-0.217 - 0.975i)T \)
17 \( 1 + (-0.980 + 0.196i)T \)
19 \( 1 + (-0.281 - 0.959i)T \)
29 \( 1 + (0.603 - 0.797i)T \)
31 \( 1 + (-0.783 + 0.621i)T \)
37 \( 1 + (0.577 - 0.816i)T \)
41 \( 1 + (-0.109 + 0.993i)T \)
43 \( 1 + (-0.621 + 0.783i)T \)
47 \( 1 + (-0.663 - 0.748i)T \)
53 \( 1 + (-0.493 + 0.869i)T \)
59 \( 1 + (0.769 + 0.638i)T \)
61 \( 1 + (0.0548 - 0.998i)T \)
67 \( 1 + (-0.646 + 0.762i)T \)
71 \( 1 + (-0.679 - 0.733i)T \)
73 \( 1 + (-0.797 + 0.603i)T \)
79 \( 1 + (0.988 - 0.153i)T \)
83 \( 1 + (-0.109 - 0.993i)T \)
89 \( 1 + (-0.909 - 0.415i)T \)
97 \( 1 + (0.558 - 0.829i)T \)
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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

−18.454861929459144819532532917120, −17.71915202691271134746638936173, −16.71141773988667762625308877690, −16.14419624753883316565363882495, −15.70341504083645892901944596169, −15.00501775920212752876038403572, −14.68692644488583658338856604638, −13.56126761517763824350916622, −13.069364827383577446480415257417, −12.505013952168717353775476548768, −11.94840058331706824389496163407, −10.56301824746097347041347014931, −9.636041454556425682493835589028, −9.26851929000702155093651713362, −8.45642179816174426365785272940, −8.056666807624833747070251765319, −7.1245671898029601447540383794, −6.70823217785102803062466394349, −5.55915435832550889794321767525, −4.796632828514425662711783563242, −4.13190734616775125528256870077, −3.53338024108573092374893453009, −2.65504508333459641786382861780, −1.59173714962974975955533756778, −0.016242702196680367750139501777, 0.95857137806715842742955568306, 2.18057362690765880825174025120, 2.93211082243803464711943134721, 3.2886987322958371773315756963, 4.10393995632023531639132419710, 4.64388659988037146824686631669, 6.078625482017039298889183182047, 6.69590132445044815344662324078, 7.542333319155639629134678610169, 8.39502900516144230001631189962, 8.91652456435578386170113195761, 9.72844039104354999396351131999, 10.4373986158597077893659878920, 11.04789626600245088627249543964, 11.6772881970011558384291037489, 12.685920298743058596499673024296, 13.21739105453360376696684969945, 13.66657962262118443298833992225, 14.437330942057636694579802992667, 15.07458350721632744730481416825, 15.763905728707867100134818396393, 16.44067929838055686329381082124, 17.69192230004868890886249410986, 18.25327848378106044953784430681, 19.014218825020819870706572004143

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