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Properties

Label 1-4017-4017.1013-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: $\chi_{4017} (1013, \cdot )$
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

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

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