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Properties

Label 1-3381-3381.848-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.999 - 0.0358i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (−0.986 − 0.162i)2-s + (0.947 + 0.320i)4-s + (−0.933 + 0.359i)5-s + (−0.882 − 0.470i)8-s + (0.979 − 0.202i)10-s + (0.742 + 0.670i)11-s + (0.0203 − 0.999i)13-s + (0.794 + 0.607i)16-s + (0.947 − 0.320i)17-s + (−0.841 + 0.540i)19-s + (−0.999 + 0.0407i)20-s + (−0.623 − 0.781i)22-s + (0.742 − 0.670i)25-s + (−0.182 + 0.983i)26-s + (−0.947 + 0.320i)29-s + ⋯
L(s)  = 1  + (−0.986 − 0.162i)2-s + (0.947 + 0.320i)4-s + (−0.933 + 0.359i)5-s + (−0.882 − 0.470i)8-s + (0.979 − 0.202i)10-s + (0.742 + 0.670i)11-s + (0.0203 − 0.999i)13-s + (0.794 + 0.607i)16-s + (0.947 − 0.320i)17-s + (−0.841 + 0.540i)19-s + (−0.999 + 0.0407i)20-s + (−0.623 − 0.781i)22-s + (0.742 − 0.670i)25-s + (−0.182 + 0.983i)26-s + (−0.947 + 0.320i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.999 - 0.0358i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (848, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.999 - 0.0358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8050997473 + 0.01442777611i\)
\(L(\frac12)\) \(\approx\) \(0.8050997473 + 0.01442777611i\)
\(L(1)\) \(\approx\) \(0.6255157476 + 0.0008416092797i\)
\(L(1)\) \(\approx\) \(0.6255157476 + 0.0008416092797i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.986 - 0.162i)T \)
5 \( 1 + (-0.933 + 0.359i)T \)
11 \( 1 + (0.742 + 0.670i)T \)
13 \( 1 + (0.0203 - 0.999i)T \)
17 \( 1 + (0.947 - 0.320i)T \)
19 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (-0.947 + 0.320i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (-0.685 - 0.728i)T \)
41 \( 1 + (0.933 - 0.359i)T \)
43 \( 1 + (-0.882 + 0.470i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.970 - 0.242i)T \)
59 \( 1 + (0.979 - 0.202i)T \)
61 \( 1 + (-0.182 - 0.983i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (0.818 + 0.574i)T \)
73 \( 1 + (-0.523 + 0.852i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (-0.301 - 0.953i)T \)
89 \( 1 + (0.917 + 0.396i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−18.90564997422336481297070407181, −18.35725180010369821782745085747, −17.143731902157012775692050260235, −16.764441753070972222265832865693, −16.387321357870059735933021057347, −15.42228989428967830042621064394, −14.92123251512984155714430793242, −14.21306476094805265970660806519, −13.18606021043160101263912853383, −12.24315543438381262484572043069, −11.63972450265521979341375866123, −11.20667483394349725069620998724, −10.37791806495406543530368022797, −9.44208387953271906989086725700, −8.83878875147797545094779128092, −8.34289430918364378805064222826, −7.48749312757498835625573330653, −6.89441738441485241333146351230, −6.08972626802050781082718839609, −5.27169854520994170549352046587, −4.08771683808698099071914561796, −3.58038618091610031837383786306, −2.419587351609849743983345206589, −1.48322635390619349817664858278, −0.62163566718889053729660202381, 0.58010044784248853690913675149, 1.586826635435899519731573927745, 2.517902564418035164557902789205, 3.49960673230360641248379499111, 3.89100669121565986229915452847, 5.19497514265738812444691106236, 6.09391751045620951973790645394, 7.04975605811512408830954311890, 7.43297254049050031852369351466, 8.19319856607710434164963423021, 8.8471845938509244490290952488, 9.707156848071964242430776156911, 10.405874382281506702868018372997, 10.97844995431186803803064447652, 11.72859291485657667890327487793, 12.45742268685442965573486317016, 12.779892778909695320078996064484, 14.37430115028691633214251062496, 14.74633092811642087333553683192, 15.48851588271616259801284547304, 16.1488525608220851476617218527, 16.80689026676814252481179071286, 17.519863001925114346519700269357, 18.17486958783428906431430013595, 18.90261680885346777820177199647

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