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Properties

Label 1-4017-4017.3392-r0-0-0
Degree $1$
Conductor $4017$
Sign $0.438 + 0.898i$
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  + (−0.153 − 0.988i)2-s + (−0.952 + 0.303i)4-s + (−0.969 − 0.243i)5-s + (−0.552 − 0.833i)7-s + (0.445 + 0.895i)8-s + (−0.0922 + 0.995i)10-s + (0.779 − 0.626i)11-s + (−0.739 + 0.673i)14-s + (0.816 − 0.577i)16-s + (0.0307 + 0.999i)17-s + (−0.650 − 0.759i)19-s + (0.998 − 0.0615i)20-s + (−0.739 − 0.673i)22-s + (−0.932 + 0.361i)23-s + (0.881 + 0.473i)25-s + ⋯
L(s)  = 1  + (−0.153 − 0.988i)2-s + (−0.952 + 0.303i)4-s + (−0.969 − 0.243i)5-s + (−0.552 − 0.833i)7-s + (0.445 + 0.895i)8-s + (−0.0922 + 0.995i)10-s + (0.779 − 0.626i)11-s + (−0.739 + 0.673i)14-s + (0.816 − 0.577i)16-s + (0.0307 + 0.999i)17-s + (−0.650 − 0.759i)19-s + (0.998 − 0.0615i)20-s + (−0.739 − 0.673i)22-s + (−0.932 + 0.361i)23-s + (0.881 + 0.473i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $0.438 + 0.898i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (3392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ 0.438 + 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02177851025 + 0.01360998993i\)
\(L(\frac12)\) \(\approx\) \(0.02177851025 + 0.01360998993i\)
\(L(1)\) \(\approx\) \(0.4963005365 - 0.3745893424i\)
\(L(1)\) \(\approx\) \(0.4963005365 - 0.3745893424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.153 - 0.988i)T \)
5 \( 1 + (-0.969 - 0.243i)T \)
7 \( 1 + (-0.552 - 0.833i)T \)
11 \( 1 + (0.779 - 0.626i)T \)
17 \( 1 + (0.0307 + 0.999i)T \)
19 \( 1 + (-0.650 - 0.759i)T \)
23 \( 1 + (-0.932 + 0.361i)T \)
29 \( 1 + (0.696 - 0.717i)T \)
31 \( 1 + (0.0922 + 0.995i)T \)
37 \( 1 + (-0.602 - 0.798i)T \)
41 \( 1 + (0.969 - 0.243i)T \)
43 \( 1 + (-0.992 - 0.122i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.332 - 0.943i)T \)
59 \( 1 + (0.552 - 0.833i)T \)
61 \( 1 + (-0.850 + 0.526i)T \)
67 \( 1 + (-0.998 - 0.0615i)T \)
71 \( 1 + (0.696 + 0.717i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (-0.273 - 0.961i)T \)
83 \( 1 + (-0.998 + 0.0615i)T \)
89 \( 1 + (-0.739 + 0.673i)T \)
97 \( 1 + (0.0307 - 0.999i)T \)
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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

−18.74638983240756599334403935664, −18.45415422064299433328641374757, −17.699913768056937840039228846436, −16.71596645671659298391231827126, −16.32858979785490026521673410187, −15.57285937546462925037187501482, −15.13001534340430297945333293577, −14.474832845512623579642203966769, −13.857923280284467035906380742908, −12.81220994490739319480830130994, −12.19760008005228975489599185774, −11.74784516896856147338398072504, −10.59432327897927139053097052232, −9.8583966019155105652591836241, −9.17330116291318462378562123081, −8.527392988153227463327813840215, −7.82668820560554685925941140926, −7.123209458742135846935905483157, −6.42734877150225801609876291931, −5.90018053956564540121049109048, −4.80342688499631106146984280467, −4.26416703977212613894727940728, −3.45476996854822833231745836519, −2.52675230425379510315299363483, −1.277128406872777766402441410708, 0.01062775022067720037582280908, 0.844356543889754565013567118755, 1.72273435340967983450073136652, 2.8521175601632285185113771499, 3.75032189237910948088807574156, 3.94588615747022371244065201810, 4.74812123251856292689766671961, 5.85086107412654881550642409184, 6.77976013056147929831198856546, 7.52892844030745785143735464066, 8.54983369333402171135015481538, 8.66282350831658547950982399409, 9.79264930552980511064225529976, 10.398157664435058104308751067064, 11.08383722230089340820378474518, 11.65666125355970283054695804431, 12.40543598563563862605261213754, 12.91386027456158846858305136510, 13.73951954308114117681904013917, 14.261651496911294915691470945387, 15.19175853733454573720242779561, 16.0312630459302109643660669689, 16.681222449928186569022230384317, 17.29363078982972143809269366652, 17.91291735986999628546171183600

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