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Properties

Label 1-4027-4027.3461-r0-0-0
Degree $1$
Conductor $4027$
Sign $0.848 - 0.529i$
Analytic cond. $18.7013$
Root an. cond. $18.7013$
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  + (0.841 − 0.540i)2-s + (−0.888 − 0.458i)3-s + (0.415 − 0.909i)4-s + (0.928 + 0.371i)5-s + (−0.995 + 0.0950i)6-s + (−0.959 + 0.281i)7-s + (−0.142 − 0.989i)8-s + (0.580 + 0.814i)9-s + (0.981 − 0.189i)10-s + (0.928 + 0.371i)11-s + (−0.786 + 0.618i)12-s + 13-s + (−0.654 + 0.755i)14-s + (−0.654 − 0.755i)15-s + (−0.654 − 0.755i)16-s + (0.580 − 0.814i)17-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.888 − 0.458i)3-s + (0.415 − 0.909i)4-s + (0.928 + 0.371i)5-s + (−0.995 + 0.0950i)6-s + (−0.959 + 0.281i)7-s + (−0.142 − 0.989i)8-s + (0.580 + 0.814i)9-s + (0.981 − 0.189i)10-s + (0.928 + 0.371i)11-s + (−0.786 + 0.618i)12-s + 13-s + (−0.654 + 0.755i)14-s + (−0.654 − 0.755i)15-s + (−0.654 − 0.755i)16-s + (0.580 − 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4027\)
Sign: $0.848 - 0.529i$
Analytic conductor: \(18.7013\)
Root analytic conductor: \(18.7013\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4027} (3461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4027,\ (0:\ ),\ 0.848 - 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.655690785 - 0.7600651719i\)
\(L(\frac12)\) \(\approx\) \(2.655690785 - 0.7600651719i\)
\(L(1)\) \(\approx\) \(1.510819410 - 0.5150929751i\)
\(L(1)\) \(\approx\) \(1.510819410 - 0.5150929751i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4027 \( 1 \)
good2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (-0.888 - 0.458i)T \)
5 \( 1 + (0.928 + 0.371i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
11 \( 1 + (0.928 + 0.371i)T \)
13 \( 1 + T \)
17 \( 1 + (0.580 - 0.814i)T \)
19 \( 1 + (0.928 + 0.371i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.327 + 0.945i)T \)
31 \( 1 + (-0.786 - 0.618i)T \)
37 \( 1 + (0.928 + 0.371i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (0.723 + 0.690i)T \)
59 \( 1 + (0.981 - 0.189i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + T \)
71 \( 1 + (0.0475 + 0.998i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (0.723 - 0.690i)T \)
89 \( 1 + (0.0475 - 0.998i)T \)
97 \( 1 + (-0.327 - 0.945i)T \)
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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

−18.19144192381095231873634242236, −17.60092883699805123425875248211, −16.899038156311287665154126552328, −16.38151562588364889093416437738, −16.116254800731567262608737380145, −15.20282930928849574332206842997, −14.371071229557812176626832373, −13.73448905852732643303885935780, −13.09140468190055724071806642447, −12.46729526837591978950668135791, −11.88759756242716833657771748827, −10.977322200081359800491433000695, −10.39113878769376752355696763315, −9.443370629139668637544817801992, −8.98981236925070139012065432157, −7.956554374855108195308019654552, −6.76796985635433819173137549572, −6.449541484099593103264317727709, −5.74390468954839285973825144871, −5.39503068123625762541586072482, −4.176314288094932058342836271274, −3.83873415212840075422400157219, −2.98355682334357469341517888436, −1.73421428774836353080853722697, −0.736387233167409895185385141659, 1.06875755179964986488136292635, 1.51747283787462064422880544359, 2.50756154673258302208114625252, 3.33413798379094832998472820781, 4.03804226427315719410177569137, 5.22922680653526245168898819863, 5.68259240190722222364788584296, 6.242731176972487906828547989264, 6.849588401800514573063264749609, 7.502161256047740606092372171200, 9.07218456365496152879613672502, 9.77147785411440864407555148160, 10.07209271354590072414162948855, 11.13839841265771536792209195418, 11.58275418770696234856836231762, 12.23971347589788337485294647981, 13.064914455023548718624893657079, 13.376805587815915034512527310869, 14.135231717134043556617506836649, 14.77302363139295812952787754118, 15.8291196075482636516069686039, 16.30717335177165915639530381942, 16.95793260637933410816019969817, 18.044503126095601057507885023334, 18.436776677718552983260212142233

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