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Properties

Label 1-4017-4017.1025-r0-0-0
Degree $1$
Conductor $4017$
Sign $0.872 + 0.489i$
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  + (−0.995 − 0.0922i)2-s + (0.982 + 0.183i)4-s + (−0.0615 + 0.998i)5-s + (−0.961 + 0.273i)7-s + (−0.961 − 0.273i)8-s + (0.153 − 0.988i)10-s + (−0.417 − 0.908i)11-s + (0.982 − 0.183i)14-s + (0.932 + 0.361i)16-s + (−0.602 + 0.798i)17-s + (−0.303 − 0.952i)19-s + (−0.243 + 0.969i)20-s + (0.332 + 0.943i)22-s + (−0.908 − 0.417i)23-s + (−0.992 − 0.122i)25-s + ⋯
L(s)  = 1  + (−0.995 − 0.0922i)2-s + (0.982 + 0.183i)4-s + (−0.0615 + 0.998i)5-s + (−0.961 + 0.273i)7-s + (−0.961 − 0.273i)8-s + (0.153 − 0.988i)10-s + (−0.417 − 0.908i)11-s + (0.982 − 0.183i)14-s + (0.932 + 0.361i)16-s + (−0.602 + 0.798i)17-s + (−0.303 − 0.952i)19-s + (−0.243 + 0.969i)20-s + (0.332 + 0.943i)22-s + (−0.908 − 0.417i)23-s + (−0.992 − 0.122i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $0.872 + 0.489i$
Analytic conductor: \(18.6548\)
Root analytic conductor: \(18.6548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4017} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4017,\ (0:\ ),\ 0.872 + 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3931066030 + 0.1027250659i\)
\(L(\frac12)\) \(\approx\) \(0.3931066030 + 0.1027250659i\)
\(L(1)\) \(\approx\) \(0.4826510961 + 0.05078295477i\)
\(L(1)\) \(\approx\) \(0.4826510961 + 0.05078295477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.995 - 0.0922i)T \)
5 \( 1 + (-0.0615 + 0.998i)T \)
7 \( 1 + (-0.961 + 0.273i)T \)
11 \( 1 + (-0.417 - 0.908i)T \)
17 \( 1 + (-0.602 + 0.798i)T \)
19 \( 1 + (-0.303 - 0.952i)T \)
23 \( 1 + (-0.908 - 0.417i)T \)
29 \( 1 + (-0.445 - 0.895i)T \)
31 \( 1 + (-0.361 + 0.932i)T \)
37 \( 1 + (-0.999 + 0.0307i)T \)
41 \( 1 + (-0.895 - 0.445i)T \)
43 \( 1 + (-0.881 + 0.473i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.952 - 0.303i)T \)
59 \( 1 + (0.717 - 0.696i)T \)
61 \( 1 + (0.992 + 0.122i)T \)
67 \( 1 + (0.243 + 0.969i)T \)
71 \( 1 + (-0.895 - 0.445i)T \)
73 \( 1 + (-0.895 - 0.445i)T \)
79 \( 1 + (0.445 + 0.895i)T \)
83 \( 1 + (0.717 + 0.696i)T \)
89 \( 1 + (0.943 + 0.332i)T \)
97 \( 1 + (0.122 + 0.992i)T \)
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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

−18.382399617264123507533704367714, −17.71522001205195018749757573286, −17.02567651971893563587233334480, −16.37979228324807182138918385854, −15.99525028664876611534081271239, −15.325951870944592947233924465, −14.53105023553011571046819280978, −13.392693509845386819025407002805, −12.93995421999851876448528454562, −12.05849395204229665231828187468, −11.701617494846972943284234989640, −10.48313940952171957681839173695, −10.024330432669251825963953351655, −9.400908067531754540771322521184, −8.80133550155842434827556896615, −7.99348005398133424647159517553, −7.33155832048869367726963751053, −6.6732461920223187619022202085, −5.79248793429552265027401173852, −5.099305146377120479392169313409, −4.06726056122967362141326858193, −3.26563182806440910570481365042, −2.13719333437702406481353931964, −1.58646261079227139974599223580, −0.358895377701540974530237309, 0.40028664985943212435291962217, 1.919910219323162982819105420069, 2.49408916908869990348914969510, 3.3187142452911815932380087628, 3.82145003241735447994129009359, 5.36558612650024931915764719707, 6.203209631434073837545393756588, 6.652811577494947868184597201083, 7.24733816878382562226550690160, 8.3822467979447962201442859874, 8.60374177286808055003133044191, 9.67986379131864524343909451429, 10.23789950914073237041270375453, 10.78176533298949240989731803343, 11.471770269411903170144087171704, 12.12932095337725241067408694123, 13.10208967904612615856538961363, 13.62420342481435407118042667390, 14.76089916215682721617205370989, 15.267189240688904404925753351622, 15.96578230612425646226881552729, 16.40569243608492534023969405486, 17.36962486547046612003283859106, 17.92595973498314620034272124454, 18.5793806220177301844592642044

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