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Properties

Label 1-4017-4017.3470-r0-0-0
Degree $1$
Conductor $4017$
Sign $0.238 - 0.971i$
Analytic cond. $18.6548$
Root an. cond. $18.6548$
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  + (0.816 − 0.577i)2-s + (0.332 − 0.943i)4-s + (−0.552 − 0.833i)5-s + (0.696 + 0.717i)7-s + (−0.273 − 0.961i)8-s + (−0.932 − 0.361i)10-s + (0.908 + 0.417i)11-s + (0.982 + 0.183i)14-s + (−0.779 − 0.626i)16-s + (−0.992 + 0.122i)17-s + (0.952 + 0.303i)19-s + (−0.969 + 0.243i)20-s + (0.982 − 0.183i)22-s + (−0.0922 + 0.995i)23-s + (−0.389 + 0.920i)25-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)2-s + (0.332 − 0.943i)4-s + (−0.552 − 0.833i)5-s + (0.696 + 0.717i)7-s + (−0.273 − 0.961i)8-s + (−0.932 − 0.361i)10-s + (0.908 + 0.417i)11-s + (0.982 + 0.183i)14-s + (−0.779 − 0.626i)16-s + (−0.992 + 0.122i)17-s + (0.952 + 0.303i)19-s + (−0.969 + 0.243i)20-s + (0.982 − 0.183i)22-s + (−0.0922 + 0.995i)23-s + (−0.389 + 0.920i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $0.238 - 0.971i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (3470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ 0.238 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.403033022 - 1.884091730i\)
\(L(\frac12)\) \(\approx\) \(2.403033022 - 1.884091730i\)
\(L(1)\) \(\approx\) \(1.571748034 - 0.7587730165i\)
\(L(1)\) \(\approx\) \(1.571748034 - 0.7587730165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.816 - 0.577i)T \)
5 \( 1 + (-0.552 - 0.833i)T \)
7 \( 1 + (0.696 + 0.717i)T \)
11 \( 1 + (0.908 + 0.417i)T \)
17 \( 1 + (-0.992 + 0.122i)T \)
19 \( 1 + (0.952 + 0.303i)T \)
23 \( 1 + (-0.0922 + 0.995i)T \)
29 \( 1 + (0.998 - 0.0615i)T \)
31 \( 1 + (0.932 - 0.361i)T \)
37 \( 1 + (-0.850 - 0.526i)T \)
41 \( 1 + (0.552 - 0.833i)T \)
43 \( 1 + (-0.881 - 0.473i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.213 + 0.976i)T \)
59 \( 1 + (-0.696 + 0.717i)T \)
61 \( 1 + (-0.602 - 0.798i)T \)
67 \( 1 + (0.969 + 0.243i)T \)
71 \( 1 + (0.998 + 0.0615i)T \)
73 \( 1 + (0.445 + 0.895i)T \)
79 \( 1 + (0.445 - 0.895i)T \)
83 \( 1 + (0.969 - 0.243i)T \)
89 \( 1 + (0.982 + 0.183i)T \)
97 \( 1 + (-0.992 - 0.122i)T \)
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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

−18.399844618323154836863279796824, −17.79780283483240723267133083621, −17.23536226479938061503019606897, −16.400872910312880141142741385335, −15.83000623997124486040771244856, −15.1070021535186519817579289080, −14.49245571440594729568099307698, −13.876640537948742280039007432602, −13.58749272813331035714124723895, −12.36321931980872455243493381279, −11.802473855946686082123484815637, −11.16184791472630011241284976888, −10.697907734184040220677401585030, −9.61118995086412621992127852938, −8.46048463781935325893806568404, −8.12028971246838300592944537349, −7.17731391572787563908565566781, −6.66125679020215042744455093796, −6.19685695920341509089890305895, −4.828014769987330348504544426368, −4.561512219732683005790833186661, −3.63416997445570842312898149158, −3.05655841182973361001872876138, −2.142039984745451106246196813825, −0.884632546516606891999457426565, 0.829179900638887804179301697264, 1.63940257689042221498397455756, 2.2867587086390839808189405427, 3.424670821555825834227926946660, 4.08098443531521585897557578928, 4.78879361722261650709061231654, 5.32653376156187352113693973867, 6.12525079629783034584833437581, 7.025072319495135918235955425722, 7.850327651626979138719895725245, 8.83072887003003481574830373644, 9.23294806260853546145464347726, 10.11555699968405187029129287649, 11.07937584363950218443436078045, 11.735446357490422387397165043811, 12.07784091239604339460308068195, 12.61910905534500457664562557350, 13.72099229610351608267111346691, 13.959634663399452717976326488050, 15.04180629404137815918806281550, 15.49009928874545668089317923972, 15.93141775620422316030078183260, 17.04497308373512872883709943828, 17.642425102154481468924068767915, 18.47641797561528749538741486853

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