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Properties

Label 1-3381-3381.131-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.394 - 0.919i$
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  + (−0.777 − 0.628i)2-s + (0.209 + 0.977i)4-s + (0.464 + 0.885i)5-s + (0.452 − 0.891i)8-s + (0.195 − 0.980i)10-s + (0.427 − 0.903i)11-s + (−0.339 + 0.940i)13-s + (−0.912 + 0.409i)16-s + (−0.951 + 0.307i)17-s + (−0.723 − 0.690i)19-s + (−0.768 + 0.639i)20-s + (−0.900 + 0.433i)22-s + (−0.568 + 0.822i)25-s + (0.855 − 0.517i)26-s + (−0.742 − 0.670i)29-s + ⋯
L(s)  = 1  + (−0.777 − 0.628i)2-s + (0.209 + 0.977i)4-s + (0.464 + 0.885i)5-s + (0.452 − 0.891i)8-s + (0.195 − 0.980i)10-s + (0.427 − 0.903i)11-s + (−0.339 + 0.940i)13-s + (−0.912 + 0.409i)16-s + (−0.951 + 0.307i)17-s + (−0.723 − 0.690i)19-s + (−0.768 + 0.639i)20-s + (−0.900 + 0.433i)22-s + (−0.568 + 0.822i)25-s + (0.855 − 0.517i)26-s + (−0.742 − 0.670i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.394 - 0.919i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.394 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3378033448 - 0.5124660568i\)
\(L(\frac12)\) \(\approx\) \(0.3378033448 - 0.5124660568i\)
\(L(1)\) \(\approx\) \(0.6797703233 - 0.1173442906i\)
\(L(1)\) \(\approx\) \(0.6797703233 - 0.1173442906i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.777 - 0.628i)T \)
5 \( 1 + (0.464 + 0.885i)T \)
11 \( 1 + (0.427 - 0.903i)T \)
13 \( 1 + (-0.339 + 0.940i)T \)
17 \( 1 + (-0.951 + 0.307i)T \)
19 \( 1 + (-0.723 - 0.690i)T \)
29 \( 1 + (-0.742 - 0.670i)T \)
31 \( 1 + (0.995 + 0.0950i)T \)
37 \( 1 + (0.704 - 0.709i)T \)
41 \( 1 + (-0.999 - 0.0407i)T \)
43 \( 1 + (-0.452 - 0.891i)T \)
47 \( 1 + (-0.733 + 0.680i)T \)
53 \( 1 + (0.476 + 0.879i)T \)
59 \( 1 + (-0.195 + 0.980i)T \)
61 \( 1 + (0.876 - 0.482i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (-0.557 - 0.830i)T \)
73 \( 1 + (-0.894 - 0.446i)T \)
79 \( 1 + (0.0475 - 0.998i)T \)
83 \( 1 + (0.882 - 0.470i)T \)
89 \( 1 + (0.128 - 0.991i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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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

−18.92288181374463811345809497881, −18.08252061089194747474056550565, −17.58346404933670053982469942621, −17.03943637112154008081467570114, −16.45045209499633538406641958186, −15.66677580392541735231898089766, −14.97087354904834395837712497464, −14.48956980624344813708782214019, −13.39368434175278954201371912564, −12.942748871804661826592877324633, −12.02374789147305334806277743641, −11.263885732905949670667884338564, −10.17607474556778084791608267640, −9.887176123797239987371688715214, −9.11608646894161439601599933530, −8.350324305708888602688367355970, −7.899734180944570935832755760870, −6.79057684444065843462196700998, −6.37764490736470632107398104498, −5.289377362633725235133321282926, −4.91280788722484455604781744724, −3.97925613866501608928697571702, −2.51675678726948688963804363267, −1.80415807328193502053995028735, −0.978938858782908180489472234090, 0.25326138537311725073748878517, 1.61064015737436092076708036486, 2.26094338320454534306932306220, 2.96620575379016344320439595190, 3.868639819065961222536776952629, 4.55595381345034379172183213690, 5.9559183309676417219947235585, 6.599777005645665905326467258637, 7.13298088170553687397053390573, 8.10806964498985863248959239064, 8.93003322837636043683286631331, 9.35822585178916004481263226758, 10.270126974640937767440158156055, 10.84794433634917036469595000386, 11.47804432102330967951471863768, 11.96306934955127746667871346544, 13.23650557201577318715804736438, 13.46914528475886231543978988510, 14.41789586821182158902235261833, 15.186928353983246005662674693, 15.9723418467700164476432342067, 16.88191928995338852911393756990, 17.27244048170161989838398405854, 17.9608275395985172407203613538, 18.7866774710899386154180979839

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