↑

Properties

Label 1-3381-3381.1073-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.888 - 0.459i$
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  + (0.999 − 0.0135i)2-s + (0.999 − 0.0271i)4-s + (0.288 − 0.957i)5-s + (0.999 − 0.0407i)8-s + (0.275 − 0.961i)10-s + (0.894 + 0.446i)11-s + (0.794 + 0.607i)13-s + (0.998 − 0.0543i)16-s + (−0.476 − 0.879i)17-s + (−0.0475 + 0.998i)19-s + (0.262 − 0.965i)20-s + (0.900 + 0.433i)22-s + (−0.833 − 0.552i)25-s + (0.802 + 0.596i)26-s + (0.523 − 0.852i)29-s + ⋯
L(s)  = 1  + (0.999 − 0.0135i)2-s + (0.999 − 0.0271i)4-s + (0.288 − 0.957i)5-s + (0.999 − 0.0407i)8-s + (0.275 − 0.961i)10-s + (0.894 + 0.446i)11-s + (0.794 + 0.607i)13-s + (0.998 − 0.0543i)16-s + (−0.476 − 0.879i)17-s + (−0.0475 + 0.998i)19-s + (0.262 − 0.965i)20-s + (0.900 + 0.433i)22-s + (−0.833 − 0.552i)25-s + (0.802 + 0.596i)26-s + (0.523 − 0.852i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.888 - 0.459i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.888 - 0.459i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.278674774 - 1.040435414i\)
\(L(\frac12)\) \(\approx\) \(4.278674774 - 1.040435414i\)
\(L(1)\) \(\approx\) \(2.331072419 - 0.3272924604i\)
\(L(1)\) \(\approx\) \(2.331072419 - 0.3272924604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.999 - 0.0135i)T \)
5 \( 1 + (0.288 - 0.957i)T \)
11 \( 1 + (0.894 + 0.446i)T \)
13 \( 1 + (0.794 + 0.607i)T \)
17 \( 1 + (-0.476 - 0.879i)T \)
19 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (0.523 - 0.852i)T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + (-0.440 + 0.897i)T \)
41 \( 1 + (-0.685 - 0.728i)T \)
43 \( 1 + (0.999 + 0.0407i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (0.855 + 0.517i)T \)
59 \( 1 + (0.275 - 0.961i)T \)
61 \( 1 + (0.115 + 0.993i)T \)
67 \( 1 + (-0.723 + 0.690i)T \)
71 \( 1 + (-0.742 - 0.670i)T \)
73 \( 1 + (0.644 + 0.764i)T \)
79 \( 1 + (0.995 - 0.0950i)T \)
83 \( 1 + (-0.933 + 0.359i)T \)
89 \( 1 + (-0.0339 - 0.999i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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

−19.2135293970272811757689547120, −17.98907396116623416859734159263, −17.59213258413414618681423316236, −16.665340217758676204020265607066, −15.90363631015870807343066887218, −15.18669737980195540310196864337, −14.75791862345959513837751666013, −13.89463697498464965775841910679, −13.52131197351272294942346158691, −12.734892094557328536237846087309, −11.89599000079823480803524200372, −11.1142306321620198120933597277, −10.76550942283370301490845318632, −10.0017671457782823628432987723, −8.88690002422989471826717130222, −8.14597275885006503525677029795, −7.10786631078575188048195235281, −6.53039118884055716825775688455, −6.06859300872905825752039786455, −5.21827975215639411547089989705, −4.23807179478960277031396475556, −3.50437366237484735071528661750, −2.93785975713390081999943284374, −2.03050041486769738574436735810, −1.11747756502625627240975185526, 1.05651342473648151774543392791, 1.71203781511080463463789020484, 2.57081614352538934941703745487, 3.70826525010025316905293220599, 4.31553119039325510348708320827, 4.8699000388752788353722026197, 5.832982124851141613916050822321, 6.37735135617815777872686097126, 7.124938618396602299278051733989, 8.12478579104516931624726411133, 8.81945491951877653494425178645, 9.69280032789274114955258423285, 10.333552396538714204113311928954, 11.51855471914584041956617703136, 11.83164210186030502975624211565, 12.48577250533923097650767527181, 13.35479297090359888144771890628, 13.823185778143451039739328837800, 14.37976187875527915875385965102, 15.390620818571147766836984629666, 15.89652601178002582304936803738, 16.62720519716269211121280146257, 17.12381687478648957604036432870, 17.947870483510068282562377159978, 19.02403407879005231149520584725

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