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Properties

Label 1-3381-3381.1154-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.895 - 0.445i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.895 - 0.445i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.895 - 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3146524471 - 1.338942746i\)
\(L(\frac12)\) \(\approx\) \(0.3146524471 - 1.338942746i\)
\(L(1)\) \(\approx\) \(0.7327562904 - 0.6199818863i\)
\(L(1)\) \(\approx\) \(0.7327562904 - 0.6199818863i\)

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

−19.07237891043246345374107272635, −18.155758545877635578271803349921, −17.77251880486973259654445122566, −17.133844561332202951870013124184, −16.207307182258197820496406007146, −15.72219181227946709585161831741, −14.89066834963585281996386606556, −14.49500855583132026804683955920, −13.65660218206229669190154925236, −13.2855491251908415930878870704, −12.11660725393357654285863057598, −11.23043363133988143196059384498, −10.72120810656408732602124091746, −9.791111444556583744033661993940, −9.04552088767994395807926493547, −8.58592486518536319377885285467, −7.519080488699909059498742867588, −6.88469461556916110487531051517, −6.37922824648850729182362084943, −5.80286563959931806651500056605, −4.568035970513604310448887788700, −4.05977249894744086071298064149, −3.16437686171418717914710028392, −2.03101613089587509102311263835, −1.00099597640796621264883422542, 0.53426338485677475946616739633, 1.35665085985625064645110187380, 2.020473591886079945929530702225, 3.18084568655462542765652269034, 3.928004960382365588020767053073, 4.50689545464551871873400314314, 5.4003248353976597948485350842, 6.13754584952779393461031223270, 7.264183818799484713329109737772, 8.37761570581272426190077836167, 8.56412911644354305281605361219, 9.430952960421534481957686668370, 9.99879461504168771920067422068, 10.911990181958559584221922382215, 11.61610057673283879709364324386, 12.12657979675416751447776347935, 12.96126888085677480704082623069, 13.36772926338209067396399777300, 14.16085503602770829191059060486, 14.93060357051782604451855849397, 16.07086501021118015377381321179, 16.47519887487097539114295464836, 17.3866515373483768713410130226, 17.800691271269566854882618686, 18.617943792717952819346414512189

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