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Properties

Label 1-4017-4017.2066-r0-0-0
Degree $1$
Conductor $4017$
Sign $0.629 + 0.776i$
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  + (0.332 − 0.943i)2-s + (−0.779 − 0.626i)4-s + (0.389 − 0.920i)5-s + (0.0307 − 0.999i)7-s + (−0.850 + 0.526i)8-s + (−0.739 − 0.673i)10-s + (−0.650 − 0.759i)11-s + (−0.932 − 0.361i)14-s + (0.213 + 0.976i)16-s + (−0.969 + 0.243i)17-s + (−0.816 − 0.577i)19-s + (−0.881 + 0.473i)20-s + (−0.932 + 0.361i)22-s + (0.982 + 0.183i)23-s + (−0.696 − 0.717i)25-s + ⋯
L(s)  = 1  + (0.332 − 0.943i)2-s + (−0.779 − 0.626i)4-s + (0.389 − 0.920i)5-s + (0.0307 − 0.999i)7-s + (−0.850 + 0.526i)8-s + (−0.739 − 0.673i)10-s + (−0.650 − 0.759i)11-s + (−0.932 − 0.361i)14-s + (0.213 + 0.976i)16-s + (−0.969 + 0.243i)17-s + (−0.816 − 0.577i)19-s + (−0.881 + 0.473i)20-s + (−0.932 + 0.361i)22-s + (0.982 + 0.183i)23-s + (−0.696 − 0.717i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $0.629 + 0.776i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (2066, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ 0.629 + 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4668504405 - 0.2225457767i\)
\(L(\frac12)\) \(\approx\) \(-0.4668504405 - 0.2225457767i\)
\(L(1)\) \(\approx\) \(0.5432908104 - 0.7578435748i\)
\(L(1)\) \(\approx\) \(0.5432908104 - 0.7578435748i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.332 - 0.943i)T \)
5 \( 1 + (0.389 - 0.920i)T \)
7 \( 1 + (0.0307 - 0.999i)T \)
11 \( 1 + (-0.650 - 0.759i)T \)
17 \( 1 + (-0.969 + 0.243i)T \)
19 \( 1 + (-0.816 - 0.577i)T \)
23 \( 1 + (0.982 + 0.183i)T \)
29 \( 1 + (-0.992 + 0.122i)T \)
31 \( 1 + (0.739 - 0.673i)T \)
37 \( 1 + (0.445 + 0.895i)T \)
41 \( 1 + (-0.389 - 0.920i)T \)
43 \( 1 + (-0.552 - 0.833i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.908 + 0.417i)T \)
59 \( 1 + (-0.0307 - 0.999i)T \)
61 \( 1 + (-0.273 + 0.961i)T \)
67 \( 1 + (0.881 + 0.473i)T \)
71 \( 1 + (-0.992 - 0.122i)T \)
73 \( 1 + (-0.602 + 0.798i)T \)
79 \( 1 + (-0.602 - 0.798i)T \)
83 \( 1 + (0.881 - 0.473i)T \)
89 \( 1 + (-0.932 - 0.361i)T \)
97 \( 1 + (-0.969 - 0.243i)T \)
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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

−18.73054921080288002532803269684, −18.22526400491424097829342597368, −17.768297865242149461722104213971, −17.050466329590894732756648601230, −16.21638747951833037183293568751, −15.38704274753304862917922023854, −15.044726686947893854510120862347, −14.58604560466366570887486313840, −13.67063227597737425223470039764, −12.989665843981763017906894102711, −12.50789615235993454573765161523, −11.550136198742177462629593936112, −10.81823808088412704609353065087, −9.88577840816932240405448333727, −9.294377255001930078937821393426, −8.51649603983681902432414950611, −7.80259429389468881353786635028, −6.97167001363149970570176329653, −6.4631821913275521757375805964, −5.75237521428757578300211048827, −5.02945232880694325622582797889, −4.332876345638094034866231865206, −3.24125166666440376943420853487, −2.59056671656483577991302986865, −1.82943829137003918354805259011, 0.138854542590667557489448043274, 0.94958727902292611045614038310, 1.80674357928395792879553630150, 2.64063612860668514821416182545, 3.54017229809629537122997605533, 4.4093539030769190611317040411, 4.79472501645887065348376071151, 5.69314298043738979592795246893, 6.39309954430947928424003214938, 7.45212562997531392139871341669, 8.49244618796426174820836105580, 8.842804981492527659423664719406, 9.72960478016591716489581120267, 10.40187013542829504513516790479, 11.060049508585009334176077308775, 11.53850124872786497293940604639, 12.67406465742156601188097545082, 13.13810500578204755437085158537, 13.49264562374869538086378896647, 14.141110890503223300398946677942, 15.15591416396313899258661192271, 15.742261034768448426164798926773, 16.86334901236002857997852496009, 17.181561161225845215966168617840, 17.84147519759211076749297567806

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