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Properties

Label 1-4017-4017.1715-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: $\chi_{4017} (1715, \cdot )$
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

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

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