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Properties

Label 1-4017-4017.2183-r0-0-0
Degree $1$
Conductor $4017$
Sign $-0.808 + 0.588i$
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  + (−0.779 + 0.626i)2-s + (0.213 − 0.976i)4-s + (0.696 − 0.717i)5-s + (0.998 − 0.0615i)7-s + (0.445 + 0.895i)8-s + (−0.0922 + 0.995i)10-s + (0.153 + 0.988i)11-s + (−0.739 + 0.673i)14-s + (−0.908 − 0.417i)16-s + (−0.881 − 0.473i)17-s + (−0.332 + 0.943i)19-s + (−0.552 − 0.833i)20-s + (−0.739 − 0.673i)22-s + (−0.932 + 0.361i)23-s + (−0.0307 − 0.999i)25-s + ⋯
L(s)  = 1  + (−0.779 + 0.626i)2-s + (0.213 − 0.976i)4-s + (0.696 − 0.717i)5-s + (0.998 − 0.0615i)7-s + (0.445 + 0.895i)8-s + (−0.0922 + 0.995i)10-s + (0.153 + 0.988i)11-s + (−0.739 + 0.673i)14-s + (−0.908 − 0.417i)16-s + (−0.881 − 0.473i)17-s + (−0.332 + 0.943i)19-s + (−0.552 − 0.833i)20-s + (−0.739 − 0.673i)22-s + (−0.932 + 0.361i)23-s + (−0.0307 − 0.999i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $-0.808 + 0.588i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (2183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ -0.808 + 0.588i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2166066771 + 0.6651584957i\)
\(L(\frac12)\) \(\approx\) \(0.2166066771 + 0.6651584957i\)
\(L(1)\) \(\approx\) \(0.7521707034 + 0.2045940267i\)
\(L(1)\) \(\approx\) \(0.7521707034 + 0.2045940267i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.779 + 0.626i)T \)
5 \( 1 + (0.696 - 0.717i)T \)
7 \( 1 + (0.998 - 0.0615i)T \)
11 \( 1 + (0.153 + 0.988i)T \)
17 \( 1 + (-0.881 - 0.473i)T \)
19 \( 1 + (-0.332 + 0.943i)T \)
23 \( 1 + (-0.932 + 0.361i)T \)
29 \( 1 + (-0.969 - 0.243i)T \)
31 \( 1 + (0.0922 + 0.995i)T \)
37 \( 1 + (-0.602 - 0.798i)T \)
41 \( 1 + (-0.696 - 0.717i)T \)
43 \( 1 + (0.389 + 0.920i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.650 + 0.759i)T \)
59 \( 1 + (-0.998 - 0.0615i)T \)
61 \( 1 + (-0.850 + 0.526i)T \)
67 \( 1 + (0.552 - 0.833i)T \)
71 \( 1 + (-0.969 + 0.243i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (-0.273 - 0.961i)T \)
83 \( 1 + (0.552 + 0.833i)T \)
89 \( 1 + (-0.739 + 0.673i)T \)
97 \( 1 + (-0.881 + 0.473i)T \)
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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

−18.3732777709174796664089564096, −17.556424415937644597576056783478, −17.14038145751548808659634418071, −16.48254107709705946080426246797, −15.36823349126824210379776308860, −14.96177088728594934808563318658, −13.77853619589340707241590481780, −13.60027070077188948815014643708, −12.66992614178616414742587709944, −11.58087195447311540454632483076, −11.31331946866223800208369097650, −10.60967057842892984858232153713, −10.08931331494117727959247818400, −9.063559331003102978029434749838, −8.65414085617878751947126046652, −7.89601722926164332218203793443, −7.0625462958509906642710523074, −6.3704727667470214961391115317, −5.56433828119433059291529000177, −4.488374000498899664677136430376, −3.711110749240065180243120872493, −2.806332043464201812685574145130, −2.07892726776432395384352486144, −1.5371440911002811099127077666, −0.23537959532570050305613868263, 1.2451312335986283023607944850, 1.78606476170477953912016928745, 2.38217593665752148900641625860, 4.11841738438033624675527290628, 4.685762800527527186088190925982, 5.45850337973009986800157698267, 6.02703255968480808588436803789, 6.984337802347100178387088032405, 7.64320342629943642703750874172, 8.32175009054644342996602507564, 9.05810825629406489632208972764, 9.53214681811064899064595812509, 10.37487100673295075382125140082, 10.90254386774048858187900988398, 11.92211004499129115735328912863, 12.4638134350082935172684974750, 13.58151260529476225743349898147, 14.076588748809138681066389850880, 14.725887143508882808698247154799, 15.467347760263725884377251020758, 16.11378305991133490745825350806, 16.909405909013333492813604620477, 17.42673285148121667151538130192, 17.91909651183203752981276649862, 18.38279041199587065453189634389

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