↑

Properties

Label 1-3381-3381.1202-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.821 + 0.570i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.534 − 0.844i)2-s + (−0.427 + 0.903i)4-s + (0.938 + 0.346i)5-s + (0.992 − 0.122i)8-s + (−0.209 − 0.977i)10-s + (0.942 + 0.333i)11-s + (0.377 − 0.925i)13-s + (−0.634 − 0.773i)16-s + (−0.568 + 0.822i)17-s + (−0.928 + 0.371i)19-s + (−0.714 + 0.699i)20-s + (−0.222 − 0.974i)22-s + (0.760 + 0.649i)25-s + (−0.984 + 0.175i)26-s + (−0.996 − 0.0815i)29-s + ⋯
L(s)  = 1  + (−0.534 − 0.844i)2-s + (−0.427 + 0.903i)4-s + (0.938 + 0.346i)5-s + (0.992 − 0.122i)8-s + (−0.209 − 0.977i)10-s + (0.942 + 0.333i)11-s + (0.377 − 0.925i)13-s + (−0.634 − 0.773i)16-s + (−0.568 + 0.822i)17-s + (−0.928 + 0.371i)19-s + (−0.714 + 0.699i)20-s + (−0.222 − 0.974i)22-s + (0.760 + 0.649i)25-s + (−0.984 + 0.175i)26-s + (−0.996 − 0.0815i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.821 + 0.570i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.821 + 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.240689111 + 0.3885148972i\)
\(L(\frac12)\) \(\approx\) \(1.240689111 + 0.3885148972i\)
\(L(1)\) \(\approx\) \(0.9231972185 - 0.1304651758i\)
\(L(1)\) \(\approx\) \(0.9231972185 - 0.1304651758i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.534 - 0.844i)T \)
5 \( 1 + (0.938 + 0.346i)T \)
11 \( 1 + (0.942 + 0.333i)T \)
13 \( 1 + (0.377 - 0.925i)T \)
17 \( 1 + (-0.568 + 0.822i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (-0.996 - 0.0815i)T \)
31 \( 1 + (-0.0475 - 0.998i)T \)
37 \( 1 + (0.665 + 0.746i)T \)
41 \( 1 + (-0.768 + 0.639i)T \)
43 \( 1 + (-0.992 - 0.122i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (0.833 + 0.552i)T \)
59 \( 1 + (0.209 + 0.977i)T \)
61 \( 1 + (-0.644 + 0.764i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (0.591 - 0.806i)T \)
73 \( 1 + (0.00679 + 0.999i)T \)
79 \( 1 + (0.723 + 0.690i)T \)
83 \( 1 + (-0.452 + 0.891i)T \)
89 \( 1 + (0.810 - 0.585i)T \)
97 \( 1 + (-0.415 + 0.909i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.55360736402966201435680518061, −17.96022857102728226086204570792, −17.29203192578921519282704888800, −16.6508423703102804401967749829, −16.29486379077699044144690291553, −15.36963103821280801316182871307, −14.52549827451883758285596443337, −14.05415334630873948351620374586, −13.42335581163606137302257442061, −12.73196815662668273232513555321, −11.53577954893393892838491632323, −10.98584747184931887067455353246, −10.042326075938583947258013787536, −9.33071928201577051721746854653, −8.908742877019248500025777138291, −8.34235233416829535869839215523, −7.09288080031620687532913318883, −6.617355684781991794521076701308, −6.05811552145171987025947310832, −5.12241048963412453160746481492, −4.54217823696123609735831035986, −3.554763595862996240304227021564, −2.08430101601882007112273702818, −1.62942776369561288421831206930, −0.482683829293653364461290861307, 1.08388185702885689898845698701, 1.8365319323974582224194163041, 2.46997952369690528313501544839, 3.48526517562314375807934607004, 4.086603261467651797556469500022, 5.09124193420550664672172244911, 6.12866219127303955565310376050, 6.65329322690815223228618833828, 7.70584651978737076482772922448, 8.46274907468546441065733901197, 9.11307780624709842768396403119, 9.90252030674634590440284711827, 10.342625953905568110289759050341, 11.10707600632509992944526511950, 11.7243232167251610926287139548, 12.7526326351696787855657229283, 13.13089554715818781184306336932, 13.76981477737962147934496731798, 14.87310782938549588765756197896, 15.14293339522012700612296502263, 16.63431191370902088586493647440, 16.98136961354205666365146138152, 17.5374188350650712440493070790, 18.38966133630105503935668646390, 18.64791104063981013335427600132

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