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Properties

Label 1-3381-3381.86-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.520 - 0.853i$
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  + (0.942 − 0.333i)2-s + (0.777 − 0.628i)4-s + (0.855 − 0.517i)5-s + (0.523 − 0.852i)8-s + (0.634 − 0.773i)10-s + (0.534 − 0.844i)11-s + (−0.818 − 0.574i)13-s + (0.209 − 0.977i)16-s + (0.155 − 0.987i)17-s + (−0.928 + 0.371i)19-s + (0.339 − 0.940i)20-s + (0.222 − 0.974i)22-s + (0.464 − 0.885i)25-s + (−0.963 − 0.268i)26-s + (0.933 − 0.359i)29-s + ⋯
L(s)  = 1  + (0.942 − 0.333i)2-s + (0.777 − 0.628i)4-s + (0.855 − 0.517i)5-s + (0.523 − 0.852i)8-s + (0.634 − 0.773i)10-s + (0.534 − 0.844i)11-s + (−0.818 − 0.574i)13-s + (0.209 − 0.977i)16-s + (0.155 − 0.987i)17-s + (−0.928 + 0.371i)19-s + (0.339 − 0.940i)20-s + (0.222 − 0.974i)22-s + (0.464 − 0.885i)25-s + (−0.963 − 0.268i)26-s + (0.933 − 0.359i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.520 - 0.853i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.520 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.864933307 - 3.321458550i\)
\(L(\frac12)\) \(\approx\) \(1.864933307 - 3.321458550i\)
\(L(1)\) \(\approx\) \(1.855603916 - 1.097886409i\)
\(L(1)\) \(\approx\) \(1.855603916 - 1.097886409i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.942 - 0.333i)T \)
5 \( 1 + (0.855 - 0.517i)T \)
11 \( 1 + (0.534 - 0.844i)T \)
13 \( 1 + (-0.818 - 0.574i)T \)
17 \( 1 + (0.155 - 0.987i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (0.933 - 0.359i)T \)
31 \( 1 + (0.0475 + 0.998i)T \)
37 \( 1 + (0.923 + 0.384i)T \)
41 \( 1 + (-0.0203 - 0.999i)T \)
43 \( 1 + (0.523 + 0.852i)T \)
47 \( 1 + (-0.365 + 0.930i)T \)
53 \( 1 + (0.511 + 0.859i)T \)
59 \( 1 + (0.634 - 0.773i)T \)
61 \( 1 + (0.248 - 0.968i)T \)
67 \( 1 + (-0.981 + 0.189i)T \)
71 \( 1 + (-0.882 + 0.470i)T \)
73 \( 1 + (-0.973 + 0.229i)T \)
79 \( 1 + (-0.723 - 0.690i)T \)
83 \( 1 + (0.970 + 0.242i)T \)
89 \( 1 + (-0.751 - 0.659i)T \)
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

−19.33877884429459577452401720889, −18.11703716599721431446026372137, −17.50176310399777437077048164518, −16.92901350118913097486455660962, −16.40042631318869353857512313006, −15.17551079654831324071119452989, −14.79769000052456004706179070876, −14.43057950553071202899670594763, −13.42948913909703885816544964469, −13.00297999481921915711662745339, −12.171946629607728969415973710963, −11.58234738362302585060686540716, −10.625967322377480345360814371094, −10.06887153842770710879427745546, −9.19299940886180601511786860914, −8.31483427519883317643535283966, −7.299695775834711339209567016179, −6.78973384789653829814361476135, −6.16511335340369599650371782277, −5.44140378189653741944188166303, −4.47090744458820678231978564518, −4.03833967199226866100086454292, −2.84198728688172431266528908471, −2.22728724511693956587911850708, −1.55215104918887732948977928881, 0.731110188625630950344433680150, 1.52148642051080318375230202737, 2.577729690750998774343308329331, 3.01371096280161863411458148234, 4.201005961359710884698361439924, 4.809341260931491967016979159169, 5.57264722905997419212690181171, 6.1662744342139912611352661458, 6.863622817358672524866990574415, 7.8662502276271287934411608171, 8.795085292257301810594258617977, 9.59797310052834330067294478775, 10.22979280695969147051491683987, 10.91807749801132159098299934318, 11.819474872157020726201390991187, 12.38822750103784023414209263397, 13.03489348947079569561675119459, 13.73647587561692911006508443778, 14.312168011300194438346583320945, 14.81223253616257963837895797762, 15.93259267992294612280693701597, 16.317430027475201130305649992112, 17.184051846208429386828268410210, 17.76453966154974624496834955798, 18.83788512649424930414013659474

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