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Properties

Label 1-4017-4017.2456-r0-0-0
Degree $1$
Conductor $4017$
Sign $0.0885 + 0.996i$
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  + (0.881 − 0.473i)2-s + (0.552 − 0.833i)4-s + (0.153 + 0.988i)5-s + (−0.816 + 0.577i)7-s + (0.0922 − 0.995i)8-s + (0.602 + 0.798i)10-s + (0.0307 − 0.999i)11-s + (−0.445 + 0.895i)14-s + (−0.389 − 0.920i)16-s + (−0.213 + 0.976i)17-s + (−0.969 − 0.243i)19-s + (0.908 + 0.417i)20-s + (−0.445 − 0.895i)22-s + (0.850 − 0.526i)23-s + (−0.952 + 0.303i)25-s + ⋯
L(s)  = 1  + (0.881 − 0.473i)2-s + (0.552 − 0.833i)4-s + (0.153 + 0.988i)5-s + (−0.816 + 0.577i)7-s + (0.0922 − 0.995i)8-s + (0.602 + 0.798i)10-s + (0.0307 − 0.999i)11-s + (−0.445 + 0.895i)14-s + (−0.389 − 0.920i)16-s + (−0.213 + 0.976i)17-s + (−0.969 − 0.243i)19-s + (0.908 + 0.417i)20-s + (−0.445 − 0.895i)22-s + (0.850 − 0.526i)23-s + (−0.952 + 0.303i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0885 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0885 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $0.0885 + 0.996i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (2456, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ 0.0885 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.165107586 + 1.066169908i\)
\(L(\frac12)\) \(\approx\) \(1.165107586 + 1.066169908i\)
\(L(1)\) \(\approx\) \(1.410727444 - 0.06726229227i\)
\(L(1)\) \(\approx\) \(1.410727444 - 0.06726229227i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.881 - 0.473i)T \)
5 \( 1 + (0.153 + 0.988i)T \)
7 \( 1 + (-0.816 + 0.577i)T \)
11 \( 1 + (0.0307 - 0.999i)T \)
17 \( 1 + (-0.213 + 0.976i)T \)
19 \( 1 + (-0.969 - 0.243i)T \)
23 \( 1 + (0.850 - 0.526i)T \)
29 \( 1 + (0.779 - 0.626i)T \)
31 \( 1 + (-0.602 + 0.798i)T \)
37 \( 1 + (-0.982 + 0.183i)T \)
41 \( 1 + (-0.153 + 0.988i)T \)
43 \( 1 + (-0.650 + 0.759i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.696 + 0.717i)T \)
59 \( 1 + (0.816 + 0.577i)T \)
61 \( 1 + (0.739 + 0.673i)T \)
67 \( 1 + (-0.908 + 0.417i)T \)
71 \( 1 + (0.779 + 0.626i)T \)
73 \( 1 + (0.932 - 0.361i)T \)
79 \( 1 + (0.932 + 0.361i)T \)
83 \( 1 + (-0.908 - 0.417i)T \)
89 \( 1 + (-0.445 + 0.895i)T \)
97 \( 1 + (-0.213 - 0.976i)T \)
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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

−18.08639944648901977067294912130, −17.143508267521139358442263505400, −17.04499330470437162572041667752, −16.1536353130769888199446434225, −15.65025644415688414580798385931, −15.00032704216541872055534483661, −14.05323213952723552180665063024, −13.5299613004885834792127395309, −12.82975133640911780706520233570, −12.46823370894423995317936264103, −11.752466306759487876199598847196, −10.80677862375080480418803936032, −9.97543315277289001555862353566, −9.19278382373668244207129188095, −8.548746464964632482078785011996, −7.58709748572590228505918973291, −6.94781436058998035874193450539, −6.44220811662785189338558715478, −5.27531820677004783712281424700, −5.00611931348154311610649424226, −4.045851395370485095001404607269, −3.55149111262355790460992358261, −2.42148959793197290040297974816, −1.68835044331412908254375102676, −0.299581047266586125644453259746, 1.18955595136240589933776977777, 2.28159409606526165114469386714, 2.86197818347765781064713638520, 3.434337148564867051393860345305, 4.214407718024728262811796293383, 5.22975541627124955907217321065, 6.09134045545860604503035314832, 6.41090963766183102068003673536, 7.02393659125743217055216193978, 8.27559764755900189366590742299, 9.00840131115665064810708641565, 9.9319041692549967942685811616, 10.56966960011479986186369498662, 11.03627810716858832857209207407, 11.77967079867611024198515872863, 12.62457581692454592270561477770, 13.11585931447541531440498145011, 13.81914166971225941165536516861, 14.542309715436497092255109601443, 15.10656981256010808432943779905, 15.6661216853640208823211586514, 16.41517849289970346029586148257, 17.24345665827803942039643680850, 18.25768606761792839036599765954, 18.84628940380751055864691205354

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