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Properties

Label 1-3381-3381.1121-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.302 - 0.953i$
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  + (0.591 − 0.806i)2-s + (−0.301 − 0.953i)4-s + (0.970 − 0.242i)5-s + (−0.947 − 0.320i)8-s + (0.377 − 0.925i)10-s + (0.882 + 0.470i)11-s + (0.488 + 0.872i)13-s + (−0.818 + 0.574i)16-s + (−0.301 + 0.953i)17-s + (0.142 − 0.989i)19-s + (−0.523 − 0.852i)20-s + (0.900 − 0.433i)22-s + (0.882 − 0.470i)25-s + (0.992 + 0.122i)26-s + (0.301 − 0.953i)29-s + ⋯
L(s)  = 1  + (0.591 − 0.806i)2-s + (−0.301 − 0.953i)4-s + (0.970 − 0.242i)5-s + (−0.947 − 0.320i)8-s + (0.377 − 0.925i)10-s + (0.882 + 0.470i)11-s + (0.488 + 0.872i)13-s + (−0.818 + 0.574i)16-s + (−0.301 + 0.953i)17-s + (0.142 − 0.989i)19-s + (−0.523 − 0.852i)20-s + (0.900 − 0.433i)22-s + (0.882 − 0.470i)25-s + (0.992 + 0.122i)26-s + (0.301 − 0.953i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.302 - 0.953i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.302 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.488832545 - 1.821340264i\)
\(L(\frac12)\) \(\approx\) \(2.488832545 - 1.821340264i\)
\(L(1)\) \(\approx\) \(1.558961985 - 0.8065633984i\)
\(L(1)\) \(\approx\) \(1.558961985 - 0.8065633984i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.591 - 0.806i)T \)
5 \( 1 + (0.970 - 0.242i)T \)
11 \( 1 + (0.882 + 0.470i)T \)
13 \( 1 + (0.488 + 0.872i)T \)
17 \( 1 + (-0.301 + 0.953i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (0.301 - 0.953i)T \)
31 \( 1 + (0.841 - 0.540i)T \)
37 \( 1 + (-0.0203 + 0.999i)T \)
41 \( 1 + (-0.970 + 0.242i)T \)
43 \( 1 + (-0.947 + 0.320i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (0.986 - 0.162i)T \)
59 \( 1 + (0.377 - 0.925i)T \)
61 \( 1 + (0.992 - 0.122i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (-0.917 - 0.396i)T \)
73 \( 1 + (-0.933 - 0.359i)T \)
79 \( 1 + (0.959 - 0.281i)T \)
83 \( 1 + (-0.979 + 0.202i)T \)
89 \( 1 + (-0.714 + 0.699i)T \)
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

−18.55952833443778487772737255526, −18.16111086084349380899077890391, −17.4959358419478032058845063874, −16.776681470396813127378626624585, −16.2583639394964845102544458603, −15.45939790421976482413654764830, −14.68372905609727043175397408532, −14.06936374974881175481305026844, −13.63658097219991614242707626193, −12.92724225628438872118017678246, −12.11949095943836164275079155032, −11.448425819950351227899971915947, −10.431251021111983877883008104532, −9.781136107854245119628842937605, −8.69024808253538251760602180667, −8.54756103356905920423072240500, −7.21314179852685986175682496249, −6.81745400285683539343923739012, −5.88264133031067260021252279788, −5.526885616846784493196897800689, −4.65938717414336712171123289777, −3.5798397119513002023726252802, −3.100397451638130164923101063324, −2.06209714950334953854810171978, −0.916304553047472781910245865301, 0.95400166773757497429789250192, 1.70871712178638916264899790052, 2.30967932390788056118929712020, 3.28065737795671752433027460456, 4.29646046589083741901214092552, 4.659136208864113695195253369729, 5.6833206612708700668225665613, 6.45603055201803889284276943826, 6.7673021407318440791361185444, 8.38981635873759098101502853000, 8.9509374487482945790206562222, 9.78449574546736417721334459488, 10.08919927506913636780094516411, 11.186501518465871802140792259384, 11.65799883922419465830490182023, 12.41376561741488871536602764842, 13.39515752141700898243781014727, 13.46813854153135851946145270766, 14.388130974960504425851600658333, 14.997659930286201592192447002587, 15.73303947429759672018758125278, 16.81403422272663526474730339694, 17.39023015018433281749980668711, 17.99317657305432349405960758411, 18.9016825173731390939623380955

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