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Properties

Label 1-3381-3381.1100-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.479 - 0.877i$
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  + (−0.182 − 0.983i)2-s + (−0.933 + 0.359i)4-s + (0.0203 − 0.999i)5-s + (0.523 + 0.852i)8-s + (−0.986 + 0.162i)10-s + (−0.999 + 0.0407i)11-s + (−0.818 + 0.574i)13-s + (0.742 − 0.670i)16-s + (−0.933 − 0.359i)17-s + (0.142 + 0.989i)19-s + (0.339 + 0.940i)20-s + (0.222 + 0.974i)22-s + (−0.999 − 0.0407i)25-s + (0.714 + 0.699i)26-s + (0.933 + 0.359i)29-s + ⋯
L(s)  = 1  + (−0.182 − 0.983i)2-s + (−0.933 + 0.359i)4-s + (0.0203 − 0.999i)5-s + (0.523 + 0.852i)8-s + (−0.986 + 0.162i)10-s + (−0.999 + 0.0407i)11-s + (−0.818 + 0.574i)13-s + (0.742 − 0.670i)16-s + (−0.933 − 0.359i)17-s + (0.142 + 0.989i)19-s + (0.339 + 0.940i)20-s + (0.222 + 0.974i)22-s + (−0.999 − 0.0407i)25-s + (0.714 + 0.699i)26-s + (0.933 + 0.359i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.479 - 0.877i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.479 - 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4340143557 - 0.7313515852i\)
\(L(\frac12)\) \(\approx\) \(0.4340143557 - 0.7313515852i\)
\(L(1)\) \(\approx\) \(0.6258515722 - 0.4017820791i\)
\(L(1)\) \(\approx\) \(0.6258515722 - 0.4017820791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.182 - 0.983i)T \)
5 \( 1 + (0.0203 - 0.999i)T \)
11 \( 1 + (-0.999 + 0.0407i)T \)
13 \( 1 + (-0.818 + 0.574i)T \)
17 \( 1 + (-0.933 - 0.359i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.933 + 0.359i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (-0.794 - 0.607i)T \)
41 \( 1 + (-0.0203 + 0.999i)T \)
43 \( 1 + (0.523 - 0.852i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.488 + 0.872i)T \)
59 \( 1 + (-0.986 + 0.162i)T \)
61 \( 1 + (0.714 - 0.699i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (-0.882 - 0.470i)T \)
73 \( 1 + (0.685 - 0.728i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (0.970 - 0.242i)T \)
89 \( 1 + (0.947 + 0.320i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−19.02832248524058056727025373331, −18.0754071925236151585203313633, −17.65092630603976083346878305332, −17.221970423291776610329570622877, −16.03591406035278497558993292396, −15.5323142958446029941817041508, −15.084256164728272429580667751270, −14.38779107818084521048056966879, −13.48650459256044039137596188987, −13.2122533235140938467158170067, −12.13302577354866072533201879520, −11.17150506756171193342149998593, −10.34328593065793207472421857915, −10.03107547798407117535273946694, −9.049931986579671820318733284521, −8.184016076267770581813083073448, −7.65271938992279136503864144100, −6.83351626648277480833009549102, −6.40709383177540363310808775244, −5.391898164816679999817947371987, −4.834632725417979573116858037456, −3.89463547754008302892816756716, −2.84498427856490494780903939351, −2.22190389437869033685780235993, −0.63364315454966820026162148353, 0.429440759446852361419355812150, 1.51907463663562454018585077549, 2.23496870209434698781688567839, 3.0419988856910816307076996863, 4.07694333287977678089571776753, 4.824860642153525749723573363446, 5.16550395065252329071047094154, 6.31925235552700146841182879677, 7.50367825006479060475797442536, 8.10978661587955848852303091578, 8.84930391042135177283129314095, 9.4563190608893977700461745011, 10.17608563658966771568284964561, 10.815332124730892853340204738728, 11.74941842051160418619196042321, 12.35076015684250023044870275606, 12.73267137883786414349596746953, 13.73646568633338516572052935513, 13.98925272818704224014975831454, 15.16550881356066428561590480334, 16.04209503267876606196192994104, 16.5743991466003864842697848474, 17.44858369609735912197476120968, 17.87346414494444423388564949352, 18.71368002006342481647550734225

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